Download Quantum Probability and Decision Theory, Revisited

arXiv:quant-ph/0211104v1 18 Nov 2002
Quantum Probability and Decision Theory,
Revisited
David Wallace
Magdalen College, Oxford
(Email: [email protected])
18th November 2002
Abstract
An extended analysis is given of the program, originally suggested by
Deutsch, of solving the probability problem in the Everett interpretation
by means of decision theory. Deutsch’s own proof is discussed, and alternatives are presented which are based upon different decision theories
and upon Gleason’s Theorem. It is argued that decision theory gives Everettians most or all of what they need from ‘probability’. Contact is
made with Lewis’s Principal Principle linking subjective credence with
objective chance: an Everettian Principal Principle is formulated, and
shown to be at least as defensible as the usual Principle. Some consequences of (Everettian) quantum mechanics for decision theory itself are
also discussed.
1
Introduction
In recent work on the Everett (Many-Worlds) interpretation of quantum mechanics, it has increasingly been recognised that any version of the interpretation
worth defending will be one in which the basic formalism of quantum mechanics
is left unchanged. Properties such as the interpretation of the wave-function as
describing a multiverse of branching worlds, or the ascription of probabilities to
the branching events, must be emergent from the unitary quantum mechanics
rather than added explicitly to the mathematics. Only in this way is it possible
to save the main virtue of Everett’s approach: having an account of quantum
mechanics consistent with the last seventy years of physics, not one in which the
edifice of particle physics must be constructed afresh (Saunders 1997, p. 44).1
1 This is by no means universally recognised. Everett-type interpretations can perhaps be
divided into three types:
(i) Old-style “Many-Worlds” interpretations in which worlds are added explicitly to the quantum formalism (see, e. g. , DeWitt (1970) and Deutsch (1985), although Deutsch has
since abandoned this approach; in fact, it is hard to find any remaining defendants of
type (i) approaches).
1
Of the two main problems generally raised with Everett-type interpretations, the preferred-basis problem looks eminently solvable without changing
the formalism. The main technical tool towards achieving this has of course
been decoherence theory, which has provided powerful (albeit perhaps not conclusive) evidence that the quantum state has a de facto preferred basis and
that this basis allows us to describe the universe in terms of a branching structure of approximately classical, approximately non-interacting worlds. I have
argued elsewhere (Wallace 2001a, 2001b) that there are no purely conceptual
problems with using decoherence to solve the preferred-basis problem, and that
the inexactness of the process should give us no cause to reject it as insufficient.
The other main problem with the Everett interpretation concerns the concept of probability: given that the Everettian description of measurement is a
deterministic, branching process, how are we to reconcile that with the stochastic description of measurement used in practical applications of quantum mechanics? It has been this problem, as much as the preferred basis problem,
which has led many workers on the Everett interpretation to introduce explicit
extra structure into the mathematics of quantum theory so as to make sense of
the probability of a world as (for instance) a measure over continuously many
identical worlds. Even some proponents of the Many-Minds variant on Everett
(notably Albert and Loewer 1988 and Lockwood 1989, 1996), who arguably
have no difficulty with the preferred-basis problem, have felt forced to modify
quantum mechanics in this way.
It is useful to identify two aspects of the problem. The first might be called
the incoherence problem: how, when every outcome actually occurs, can it even
make sense to view a measurement as indeterministic? Even were this solved,
there would then remain a quantitative problem: why is that indeterminism
quantified according to the quantum probability rule (i. e. , the Born rule), and
not (for instance) some other assignment of probabilities to branches?
In my view, the incoherence problem has been essentially resolved by Saunders, building on Parfit’s reductionist approach to personal identity. (Saunders’
approach is summarised in section 3). This then leaves the quantitative problem
as the major conceptual obstacle to a satisfactory formulation of the Everett
interpretation.
Saunders himself has claimed (1998) that the quantitative problem is a nonproblem: that once we have shown the coherence of ascribing probability to
quantum splitting, we can simply postulate that the quantum weights are to be
(ii) “Many-Minds” approaches in which some intrinsic property of the mind is essential to
understanding how to reconcile indeterminateness and probability with unitary quantum mechanics (see, e. g. , Albert and Loewer (1988), Lockwood (1989, 1996), Donald
(1997), and Sudbery (2000)).
(iii) Decoherence-based approaches, such as those defended by myself (Wallace 2001a, 2001b),
Saunders (1995, 1997, 1998), Deutsch (1996, 2001), Vaidman (1998, 2001) and Zurek
(1998).
For the rest of this paper, whenever I refer to “the Everett interpretation”, I shall mean
specifically the type (iii) approaches. This is simply for brevity, and certainly isn’t meant to
imply anything about what was intended in Everett’s original (1957) paper.
2
interpreted as probabilities:
Neither is it routinely required, of a physical theory, that a proof be
given that we are entitled to interpret it in a particular way; it is anyway unclear as to what could count as a proof. Normally it is enough
that the theory can be subjected to empirical test and confirmation;
quantum mechanics can certainly be applied, on the understanding
that relations in the Hilbert-space norm count as probability . . . It
is not as though the experimenter will need to understand something more, a philosophical argument, for example, before doing an
experiment. [Saunders 1998, p. 384; emphasis his.]
Whether one accepts such a claim depends upon one’s attitude to the philosophy of probability in general. Many have claimed (e. g. , Mellor 1971) that
objective probability is simply another theoretical posit, like charge or mass,
in which case presumably all that is required to introduce probability into a
theory is a mathematical structure satisfying the Kolmogorov axioms together
with the statement that the structure is to be interpreted as probability. If this
is acceptable in classical physics, then it seems no less so in quantum mechanics.
But there is a more demanding view of probability, eloquently defended by
Lewis (1980, 1994) and more recently argued for in the quantum context by
Papineau (1996): whatever (objective) probability is, our empirical access to
it is via considerations of rationality and behaviour: it must be the case that
it is rational to use probability as a guide to action. This seems to call for
just the ‘proof’ which Saunders rejects: some argument linking rational action
to whatever entities in our theory are called ‘probabilities’. Granted, there is
no really convincing such account (to my knowledge) in classical philosophy of
probability, but there is at least some prospect that either straight frequentism
(Howson and Urbach 1989, pp. 344–347) or Lewis’s Best-Systems Analysis variant on frequentism can be made to suffice But clearly neither are applicable to
the Everett interpretation, which seems to leave it at a significant disadvantage.
In this context it is extremely interesting that David Deutsch has claimed
(Deutsch 1999) to derive the quantum probability rule from decision theory:
that is, from considerations of pure rationality. It is rather surprising how
little attention his work has received in the foundational community, though
one reason may be that it is very unclear from his paper that the Everett
interpretation is assumed from the start.2 If it is tacitly assumed that his work
refers instead to some more orthodox collapse theory, then it is easy to see that
the proof is suspect; this is the basis of the criticisms levelled at Deutsch by
Barnum et al, (2000). Their attack on Deutsch’s paper seems to have been
2 Nonetheless
it is assumed:
However, in other respects he [the rational agent] will not behave as if he believed
that stochastic processes occur. For instance if asked whether they occur he will
certainly reply ‘no’, because the non-probabilistic axioms of quantum theory
require the state to evolve in a continuous and deterministic way. [Deutsch
1999, pp. 13; emphasis his.]
3
influential in the community; however, it is at best questionable whether or not
it is valid when Everettian assumptions are made explicit.
The purpose of this paper, then, is threefold: first, to give as clear an exegesis
as possible of Deutsch’s argument, making explicit its tacit assumptions and
hidden steps, and to assess its validity; second, to place his argument in the
context of general decision theory and show how we can thereby improve upon
it, making its assumptions more plausible and its argument more transparent;
thirdly, to assess the implications of Deutsch’s proof (and my variations on it)
both for the Everett interpretation and for decision theory itself.
Since decision theory is probably unfamiliar to many people working in the
foundations of quantum mechanics, I begin (section 2) by expounding its general
principles and goals. In section 3 I explain how decision theory can be applied
to quantum-mechanical contexts, and especially to the quantum games which
feature so heavily in Deutsch’s proof.
Sections 4–7 are the core of the paper. Section 4 introduces Deutsch’s games,
and shows that with very few assumptions beyond the mathematical formalism
of quantum mechanics it is possible to prove some strong results about a rational agent’s preferences between such games. These results are then central in
the proofs of sections 5–7, which give a reconstruction of Deutsch’s proof as well
as three variants on it: one closely related to Deutsch’s proof, one based upon
axioms inspired by Savage’s (1972) axiomatization of decision theory, and one
based upon Gleason’s theorem. The last of these is only tentatively proved, and
shows that the usefulness of Gleason’s theorem in a realistic reconstruction of
probability is more questionable than has sometimes (e. g. Barnum et al 2000)
been suggested. In sections 8–9 I discuss the implications of Deutsch’s proof
and its variants: section 8 is an analysis of the extent to which the proof allows quantum probability to satisfy the rationality requirements of Lewis and
Papineau, whilst section 9, a dialogue between an Everettian and a Sceptic,
discusses the possible problems and weak points of this approach and concludes
with a summary of its implications.
2
Classical decision theory
In this section, I will give a brief exposition of classical (i. e. , standard) decision
theory: both its aims and its technical details. The latter will be relevant
because, as we will see, the quantum-mechanical derivations of probability which
form the core of the paper are in many ways rather closely modelled on the
classical ones.
2.1
The decision problem
Decision theory is concerned with the preferences of rational agents — where
“rational” is construed in a rather narrow sense. If someone were to choose to
jump into an alligator pit we would be inclined to call their choice irrational,
but from a decision-theoretic viewpoint it would simply be unusual. If an agent,
4
however, were to say that they preferred alligator pits to snake pits, snake pits
to scorpion pits, and scorpion pits to alligator pits, then decision theory would
deem their preferences irrational. Decision theory, then, is concerned with the
logical and structural constraints which rationality places on an agent’s structure
of preferences, but is not intended to come anywhere near determining those
preferences wholly.
It is also concerned, in large part, with decision-making under uncertainty.
If for any action which an agent takes they are certain what the outcome will
be, then the constraint alluded to above — that, if A is preferred to B and B to
C, then A is preferred to C — is really all that decision theory has to say about
the agent’s preferences. But when the agent has to choose between a number of
acts none of which have a perfectly predictable outcome — betting on horses,
for instance, or choosing whether or not to cross the road — then considerations
of pure rationality can place strong constraints on that agent’s preferences.
To understand this further, let us define a certain sort of decision problem.
This problem is somewhat stylised but nonetheless can be used to describe a
wide class of real-world decision problems; it is a mildly enlarged version of the
decision problem considered by Savage (1972).
The decision problem is specified by the following sets:
• A set C of consequences, to be regarded as the “atomic holistic entities
that have value to the individual” (Fishburn 1981). Typical consequences
might be receiving a thousand-euro cheque, or being hit by a bus.
• A set M of chance setups, to be regarded as situations in which a number
of possible events might occur, and where it is in general impossible for
the agent to predict with certainty which will occur. Examples might be
a rolled die (in which case there is uncertainty as to which face will be
uppermost) or a section of road over a five-minute period (in which case
there is uncertainty as to whether or not a bus will come along).
• A set S of states, to be regarded as possible, in general coarse-grained,
states of the world at some time; typical states might be that state in
which lottery ticket 31942 is drawn, or in which there is a bus coming
down the road. For each M ∈ M we can define a subset SM ⊆ S of states
which might occur as a consequence of the chance setup. (SM is taken
as consisting of mutually exclusive, jointly exhaustive states.) (We will
define an event for M as some subset of SM . The event space for M is
the set EM of events, i. e. the power set of SM .)
• A set A of acts, to be regarded as possible choices of action for the individual (usually in the face of uncertainty as to which state is to arise).
For our purposes each act can be understood as a pair f = (M, P) where
M is a chance setup and P is a function from the set SM of states consistent with M , to the set C of consequences. (Thus, performing an act
is a two-part process: the chance setup M must be allowed to run, and
P fixes the consequences for the individual of the various results of M .)
5
In a sense P represents some sort of “bet” placed on the outcome of M
(such as on which die-face is uppermost), although it need not be a formal wager: in the case of the road, for instance, one payoff scheme might
refer to crossing the road, so that P(bus in road)=being hit by bus, P(no
bus)=getting safely to other side; another might refer to choosing not to
cross.3
• A preference order ≻ on the set of acts, so that f ≻ g if and only if the
agent would prefer it that f is performed than that g is. If f and g are
acts which it is within the agent’s power to bring about, this implies that
the agent would choose f rather than g; more generally it is a hypothetical
preference: if the agent were able to choose between f and g, they would
choose f .
(In developing the axiomatics of decision theory we can simply treat acts,
states and consequences as primitives; to apply the theory to the actual world,
of course, there are many subtleties as to exactly how to carve up the world
into states, how to distinguish between states and consequences, how to analyse
an act as a pair (M, P) etc. In the quantum decision problems we analyze,
however, it will be reasonably clear what the correct analysis is.)
The decision problem, put informally, is then: how do considerations of
rationality constrain an agent’s preferences between elements of A?
2.2
Expected utility
The standard answer to the decision problem is as follows: act f is preferred
to act g iff EU (f ) > EU (g), where for any act f = (M, P) the expected utility
EU (f ) of f is defined by
X
EU (f ) =
Pr(x|M )V[P(x)],
(1)
x∈SM
and where
• Pr(x|M ), a real number between zero and one, is the conditional probability that x will obtain given that M occurs;
• P(x) is the consequence associated by f with state x ∈ SM ;
• V(c), a real number, is the value to the individual of consequence c.
(In fact, if the set SM is infinite the notion of probability must be applied to (a
certain Boolean sub-algebra of) events of M rather than states simpliciter.)
3 It is important for the development of the theory that states do not per se have value to
agents; an agent values a state’s obtaining only insofar as it is associated via an act to a good
consequence. Joyce (1999) has criticised this assumption in the general case: it is hard to see
how the state ‘the Earth is destroyed’ can be rendered innocuous by appropriate choice of
consequence! However, in this paper the states will by and large be readout states of quantum
measurement processes, for which the assumption is far more reasonable.
6
If the notions of ‘probability’ and ‘value’ are treated as primitive, then this
expected-utility rule gives a complete answer to the decision problem, and very
strongly constrains the agent’s choices between events with uncertain outcomes:
given the probabilities and the values of consequences, then the preference order amongst acts is fixed. (The question would remain: how is the rule itself
justified?)
However, it is rather unclear what these primitive notions of probability and
value actually refer to. Understanding them qualitatively is not difficult: for the
value function, to say that V(c) > V(d) is to say that the agent would prefer
to receive c than to receive d. As for probability, if ‘more likely than’ can be
treated as primitive then we can understand ‘Pr(x|M ) > Pr(y|M )’ as saying
that x is more likely than y to occur. If we do not want to treat it as primitive
then we can understand it in terms of preferences between bets: to say that
one state is more likely than another is to say that we would rather bet on the
occurrence of the first state than of the second (assuming we don’t care per se
which state occurs).
But to define the expected-utility rule we need to understand the notions
quantitatively. If the above qualitative understanding is all that constrains
them, then we could replace Pr and V by arbitrarily monotonically increasing
functions of Pr and V, for as yet we have given no meaning to ideas like c is
twice as valuable as d, or x is half as likely as y.
We might hope to find reasons outside decision theory to make quantitative
sense of V and Pr. Maybe probabilities are objective chances (see section 2.7);
maybe V measures some “moral worth” of a consequence to us (as originally
suggested by Bernoulli; see Savage (1972, pp. 91–104) for a historical discussion).
But this is not the approach taken by decision theory. The aim, instead,
is to derive some quantitative aspects of Pr and V — and, in particular, the
EUT — from the agent’s preferences between acts (not just consequences). (In
doing so, of course, we abandon any hope that the expected-utility rule always
tells us what to do in situations of uncertainty, and fall back on the idea of
decision theory as constraining our preferences rather than determining them
completely.)
It is fairly easy to see how, given a quantitative notion of value and a preference ordering on acts, we could use it to define quantitative probabilities.
Suppose c, d and e are consequences such that our agent is indifferent between
an act where he receives c just if some state in event A occurs and d otherwise,
and another where he receives e with certainty; then we define the probability
of A by
Pr(A) =df (V(e) − V(d))/(V(c) − V(d)).
(2)
In effect, this defines probability by betting: the probability of an event is the
shortest odds at which we would be prepared to bet on it.4 (This notion of
probability was first suggested by Ramsey (1931), and has been explored in
extenso since then; see, e. g. , de Finetti 1974 or Mellor 1971.)
4 To
see this more clearly, specialise to the case where V(d) = 0.
7
Conversely, if we already have a quantitative notion of probability then we
can use it to define quantitative value of consequences (this was first advocated
by Pareto; again, see Savage (1972, pp. 91–104) for a brief history). If an agent is
indifferent between receiving e with certainty, and receiving c if state A obtains
and d otherwise, then
(V(e) − V(d))/(V(c) − V(d)) =df Pr(A).
(3)
This fixes all ratios of differences of values, and hence fixes values up to a
multiplicative and an additive constant.
Plainly, in both cases there would be a need for additional assumptions to
ensure that these definitions were self-consistent and to derive the expectedutility rule. In any case, though, they are not completely satisfactory as each
takes one of Pr and V as primitive. A more satisfactory approach would generate
both from purely qualitative axioms about preference. Only one of the two would
need to be thus generated: the other could then be defined in terms of the first,
as above.
This is, in fact, possible. The structure of such an approach is as follows:
we introduce, by means of some decision-theoretic postulates, enough structure
to the qualitative orderings of either C or EM , that it is possible to prove some
representation theorem guaranteeing, as appropriate, either an effectively unique
value function, or a unique probability function.5 Then we apply the methods
above to generate either V from Pr, or vice versa.
In either case, the ultimate goal is the same: from some basic axioms of
rational preference, prove the following representation theorem:
Expected Utility Theorem (EUT): Any agent’s preferences amongst
acts determine a unique probability measure on events and a value
function, unique up to multiplicative and additive constants, such
that the preferences are completely represented by the expectedutility rule.
This result will be partly descriptive, partly normative: only by knowing the
agent’s preferences amongst some substantial subset of acts can Pr and V be
determined, but once they are determined the remaining preferences are fixed.
As such we will have a strong constraint on the actions of rational agents.
In the next few sections, we will show how the EUT can be derived in
two ways: one beginning with a derivation of V, one with Pr. As we will
see, the approach which first proves a representation theorem for probability
and then derives utility is technically rather more complex than its converse,
but foundationally is far more satisfactory. For clarity, I will begin with the
simpler utility-based approach (in section 2.4); the probability-based approach
5 Value functions are in general only unique up to overall multiplicative and additive constants; this can be seen from the expected-utility rule, from which it is clear that multiplicative
and/or additive changes to V do not affect whether EU (f ) > EU (g). We should also stress
that uniqueness here is only relative to a given observer’s act preferences. There is nothing
at all to prevent an agent preferring death to ice cream or having preferences which constrain
the probability of the sun rising tomorrow to be zero.
8
is in sections 2.5–2.6. First, though, I shall make some general remarks about
decision-theoretic axioms.
2.3
The nature of decision-theoretic axioms
To prove our representation theorems, we have to introduce a number of axioms
of decision. In specifying these, we will need to make use of the notion of a
weak ordering. Recall6 that a weak ordering is a relation (which we will always
denote ) which is:
• transitive: if x y and y z, then x z;
• total: either x y, or y x.
We will write x ≃ y whenever x y and y x, and x ≻ y whenever x y but
x≃
/y; the relation can equally be specified in terms of ≻, defining x ≃ y whenever
x≻
/y and y ≻
/x (though the axiomatization is mildly more complicated).
We will also make use of the idea of a null event : a null event N is one
to which a rational agent is completely indifferent. Formally this means that
the agent is indifferent between any two payoff schemes which differ only on N ;
informally, it means that N has probability zero, though of course this cannot
be taken as a definition.
The axioms themselves will be seen to break into two categories, which can
be described as follows (I follow Joyce’s(1999) terminology [need to check ref to
Suppes in Joyce]):
1. Axioms of pure rationality. These are intended to be immediately selfevident principles of decision-making: the rule about transitivity of preferences mentioned in section 2.1 is one of them.
2. Structure axioms. These are mathematically-inspired axioms, not nearly
so self-evident as the axioms of pure rationality: their purpose is to rule
out possibilities such as infinitesimal probability or infinitely valuable consequences.
In all versions of classical decision theory explored to date (not just in those
presented here) it is necessary to assume some fairly strong structure axioms to
get a unique representation theory. A partial justification for them as “axioms
of coherent extendability” has been advocated by Kaplan (1983) and others
(see Joyce (1999) for some comments on this approach). Nonetheless it is generally accepted that structure axioms are less satisfactory than axioms of pure
rationality, and that their use should be minimized.
Whilst it is in general fairly straightforward to distinguish between the two
sorts of axiom, I should mention one controversy. We will be assuming, as
an axiom of pure rationality, that the preference order between acts is a weak
ordering: that is, for any two acts an agent either prefers one to another or is
indifferent between them. This is questionable: it is defensible to argue that an
6 Or
take as a definition; the nomenclature in the literature is somewhat variable.
9
agent may simply have no opinion as to which act is better. Whether this is
ultimately a sustainable viewpoint depends in part on one’s theory of desires and
preferences: Lewis, for instance, regards preferences as wholly determined by
dispositions to act (Lewis 1980), in which case preference ordering is necessarily
total; Joyce (1999) criticises this view. Further investigation of this controversy
lies beyond this paper.
2.4
Defining value functions through additivity
If we wish to treat quantitative value as prior to quantitative probability (section
2.2’s first approach to the expected-utility rule) then we will need some nonprobabilistic way of understanding the idea that one consequence is twice as
valuable as another. The only one of which I am aware is composition: c is twice
as valuable as d if I would be indifferent between receiving c, and receiving d
twice.
More formally, let us assume that there is some (associative, commutative)
operation + of composition on the set of consequences, so that the consequence
c + d is to represent receiving both c and d. (We will as usual use ‘nc’ to
abbreviate ‘c + c + . . . c’ (n times)’). This strongly restricts the elements of C, of
course: many consequences, such as, “becoming president of the EU”, cannot
be received indefinitely many times.
We will define a weak ordering ≻ on any set C with a composition operation
as additive if it satisfies:
A1 There exists a “zero consequence” 0 such that c + 0 = c for all c;7
A2 c ≻ d if and only if c + e ≻ d + e for any e;
A3 Whenever c ≻ d ≻ 0, there exists some integer n such that nd ≻ c;
A4 Whenever c ≺ 0 and d ≻ 0, there exists some integer n such that c+nd ≻ 0.
A5 For any c, d where c ≻ d, and for any e, there exist integers m, n such that
nc ≻ me ≻ nd.
It is easy to prove that A1 − A5 are equivalent to the existence of a value
function V, unique up to a multiplicative constant, such that (a) V(c) > V(d)
iff c ≻ d and (b) V(c + d) = V(c) + V(d). (The proof that such a value function
implies A1 − −A5 is trivial. The essence of the proof of the converse is: pick
an arbitrary consequence c ≻ 0, assign it value 1, and put V(d) = n whenever
d ≃ nc. For negative values, put V(d) = −n whenever d + nc ≃ 0; for rational
values, put V(d) = m/n whenever nd ≃ mc. Postulates 3 and 4 ensure that all
consequences have finite value; postulate 5 handles irrational value. The details
of this proof are in the appendix.)
7 Note that the zero consequence is unrelated to the ‘null event’ of section 2.3: the latter is
an event which (informally) I am certain will not occur, whereas the former is a consequence
which I may be certain I will receive, but am indifferent about receiving.
10
A2–A5, when we understand them as applying to consequences (or acts) are
all structure axioms, and most are pretty innocuous: A3 and A4 rule out infinitely valuable consequences, and A5 rules out infinitesimally different values.
A2, however, is a very substantive assumption which isn’t true for most of us:
let c be the consequence of becoming a billionaire; let d be the consequence of
getting a wonderful, hassle-free cruise in the South Pacific; let e be the consequence of receiving 100, 000 euros. For most of us (i. e. ., assuming that the
reader isn’t already a billionaire), even a wonderful holiday is over-priced at
100, 000 euros, so e ≻ d; however, a billionaire might very well pay 100, 000
euros to guarantee a good time, so e + c ≺ d + c.
As such, the assumption that a preference ordering is additive is at best
an approximation, applicable to certain special circumstances (gambling with
small sums is an obvious example). If we do assume additivity, though, we can
state a set of axioms which imply the expected utility rule:
U0: Act availability The set A of acts consists of all acts (M, P), for some
fixed M ∈ M and for arbitrary functions P : SM → C. (Since M is fixed,
when writing acts we will drop the M and identify acts with their payoff
functions: (M, P) ≡ P.)
U1: Preference ordering There exists a weak ordering ≻ on A (this in turn
defines a weak ordering on C, by restricting ≻ to constant P and identifying
such P with their constant value.)
U2: Dominance If P1 (s) P2 (s) for all s ∈ SM , then P1 P2 .
U3: Composition There is an operation + of composition on A such that
(M, P1 ) + (M, P2 ) = (M, P1 + P2 )
(4)
(where addition on consequences is defined similarly to the weak ordering
on consequences, by restriction of + to constant acts).
U4: Act additivity The weak ordering ≻ on A is additive (that is, satisfies
A1–A5) with respect to +.
From these axioms we can deduce that:
1. The order ≻ on the set C of consequences defines a value function V, unique
up to multiplication (this is just the result proved above, of course).
2. To each event s ∈ SM there existsP
some unique real number
P Pr(s) (between
0 and 1) such that P1 ≻ P2 iff s Pr(s)V · P1 (s) > s Pr(s)V · P2 (s).
This is of course the expected-utility theorem.
Of the axioms:
• U0 is a structure axiom, concerning those acts which we can consider.
Some such structure axiom will be used throughout the paper, and its
validity will not be discussed further.
11
• U1 and U2 are axioms of pure rationality. U1 is familiar; U2 is the assertion that, if one act is guaranteed to give consequences as good as another
act, then we should not prefer the second act to the first.
• U3 and U4 encode additivity, which has been extended from consequences
to acts in order to ensure the well-definedness of the probabilities. To
understand the justification of additivity, think of acts as the placing of
bets on all the possible outcomes of some fixed chance event M : additivity
says that our preference on possible bets doesn’t depend on what bets we
have already placed, which again is implausible in general but may be a
reasonable approximation in some circumstances.
2.5
Defining utility from probability: the von NeumannMorgenstern approach
Though the derivation of EUT from additivity gives some insight into decision
theory, additivity is an unreasonably strong assumption. We therefore consider
the alternative strategy for deriving EUT sketched in section 2.2, which begins
by proving a representation theorem for probability and then goes on to define
the value function. In the next section we discuss that representation theorem;
in this section we address the question of exactly what is needed to derive
quantitative value from quantitative probability.
This question was originally answered by von Neumann and Morgenstern
(1947). They proved a result which, in modern terminology (I follow Fishburn
1981) may be expressed as follows:
Let c1 , . . . , cn be a set of possible consequences (intended to be regarded as jointly exhaustive). Define a gamble F as anPact which will
lead to consequence ci with probability PrF (i), with i PrF (i) = 1.
Define convex sums of gambles as follows: if F and G are gambles
and 0 ≤ λ ≤ 1, then λF + (1 − λ)G is the gamble which assigns probability λ PrF (i) + (1 − λ) PrG (i) to consequence ci . Then assume:
VNM0: Gamble availability A is some set of gambles, which
contains all constant gambles (that is, gambles F for which
PrF (i) = 1 for some i) and is closed under convex sums.
VNM1: Transitive preferences There exists a weak order ≻ on
the set A.
VNM2: Sure-thing principle If F ≻ G and 0 < λ < 1 then for
any gamble H, λF + (1 − λ)H ≻ λG + (1 − λ)H.
VNM3: Gamble structure axiom If F ≻ G and G ≻ H then
there exist some α, β ∈ (0, 1) such that αF + (1 − α)H ≻ G
and G ≻ βF + (1 − β)H.
Then (von Neumann and Morgenstern showed) there exists a real
function V on the set of consequences, uniquely given up to affine
transformations and such that
12
• ci ≻ cj iff V(ci ) > V(cj ). (Here constant gambles are identified
with their values.)
• If F and G are gambles, then F ≻ G iff EU (F ) ≻ EU (G),
where
X
Pr(i)V(ci ).
(5)
EU (F ) =
i
F
Of the assumptions above, VNM0 is a structure axiom analogous to U0 and
VNM1 is the usual transitive-preferences assumption. VNM2 is essentially a
statement about the meaning of probability, an example of a class of principles
called ‘Sure Thing Principles’ by Savage (1972); in words, it could be stated as
“either x or y will occur. Of the two bets I could choose to make,
I’m indifferent between them if x occurs, and I’d prefer to have made
Bet 1 if y occurs; therefore, I should go for Bet 1, as it’s a sure thing
that I’ll either prefer that to Bet 2, or at least not mind which bet
I made.”
Sure thing principles (and dominance assumptions, come to that) are more
controversial than at first they appear; see Gärdenfors and Sahlin (1988) for
a range of criticisms of them. Nonetheless we will make use of them without
further analysis.8
The other assumption, VNM3, is another structure axiom: in effect, it says
that if one bet is preferred to another, then there will exist some change to the
probabilities involved in the bets which is so small that the preference order
is unchanged. This rules out the possibility of infinitesimal value differences
between consequences.
What is shown by the von Neumann-Morgenstern approach? That if we
assume the probabilities known, assume a merely qualitative preference order
on gambles (with no need for compositions of acts or for a zero consequence), and
make some extremely reasonable-seeming assumptions about how probabilities
affect our preferences, then we can deduce the EUT. We now turn to the problem
of deducing the probabilities themselves.
2.6
Defining probability: the Savage axioms
The project of deriving quantitative probabilities from qualitative preferences
was carried out with great clarity by Savage (1972), and we will sketch his
analysis here. He first defines a notion of qualitative probability by the method
given in section 2.2: that is, event A is more probable than event B (A ≻ B)
just if, for any consequences c and d with c ≻ d, the agent would prefer (c if A
obtains and d otherwise) to (c if B obtains and d otherwise). Then he gives an
8 One reason for this is that the most convincing objections to the Sure Thing principle
concern situations where we feel we have more knowledge about one situation than about
another; as we shall see, in analysing quantum probabilities we shall assume throughout that
we have perfect knowledge.
13
axiom system which allows him to turn this qualitative notion of ‘more probable
than’ into a quantitative probability measure.
Savage’s axiom system is extremely powerful, and hence inevitably rather
complex: it consists of the following (taken directly from Savage 1972, with the
tacit axiom S0 made explicit, following Joyce 1999.)
S0: Act availability The set A of acts consists of all acts (M, P), for some
fixed M ∈ M and for arbitrary functions P : SM → C. (As such, when
writing acts we will drop the M and identify acts with their payoff functions: (M, P) ≡ P.)
S1: Preference ordering There exists a weak ordering ≻ on the set of acts.
(The ordering defines, by an obvious restriction, an ordering on consequences: define Pc and Pd as constant acts, with Pc (x) = c and Pd (x) = d
for all x; then define c ≻ d iff Pc ≻ Pd . This also allows us to compare acts
and consequences: act F1 is preferred to consequence c iff it is preferred
to the constant act Pc . )
S2: Non-triviality Not all consequences are equally valuable: that is, there
are at least 2, c and d say, such that c ≻ d.
S3: Sure-Thing Principle If two acts F1 and F2 agree with each other on
some event C ⊆ S, then whether or not F1 ≻ F2 is independent of the
actual form of F1 (and thus F2 ) on C.
S4: Dominance If {Ci } is some partition of SM into finitely many events, if
F1 and F2 are acts with F1 = c1,i on restriction to Ci and F2 = c2,i on
restriction to Ci , and if c1,i c2,i for all i, then F1 f2 . If in addition
there is at least one i with Ci non-null such that c1,i ≻ c2,i , then F1 ≻ F2 .
S5: Probability Given any two events C and D, then either C is more probable than D, or it is less probable, or the two are equiprobable.
S6: Savage’s Structure Axiom If F1 and F2 are acts with F1 ≻ f2 , and c is
an arbitrary consequence, then there exists a partition of SM into finitely
many events {Ci } such that if either or both of F1 and F2 are modified to
be equal to c on an arbitrary Ci , the preference order is unchanged.
S7: Boundedness If F1 and F2 are acts such that F1 ≻ F2 (s) for every s,
then F1 ≻ F2 (and vice versa).
Most of these axioms are axioms of pure rationality: S1, S3 and S4 we have
essentially met before, S2 guarantees that agents have some preferences (and
hence that their beliefs about probability can be manifested in their choice of
bets), and S7 says that if one act is preferred to all the consequences of another
act, it is preferred to that other act simpliciter. S5 is rather more substantive:
it says that the probability of a given event is independent of the act performed,
Pr(A|P) = Pr(A). (This is why we have to restrict, in S0, to acts within a fixed
chance setup M .) S0 is a structure axiom equivalent to U0 or VNM0. S6 is a
new and very substantive structure axiom: it effectively says that:
14
1. events are continuously divisible, so that any event can be broken into N
sub-events each of which are equally probable.
2. No event has infinitesimal probability.
3. No consequences are infinitely valuable.
4. No consequences differ in value only infinitesimally.
Savage’s derivation of the EUT then proceeds as follows. Firstly it is shown
(by virtue of S0-S5) that ‘more probable than’ is a weak ordering on the set of
events. S6 is then used to establish that there is one and only one probability
measure on the events which is compatible with this weak ordering: the method,
in essence, is to break SM into a very large group N of equiprobable events
B1 , . . . , BN , each of which must be assigned probability 1/N , and approximate
each event by some finite union of these BN . If an event contains M of the
BM it must have probability greater than M/N ; if it is contained within the
union of M ′ of them then it must have probability less than M ′ /N . Iterating
this process for successively larger N gives, in the limit, unique probabilities for
each event.
It is perhaps worth recalling the fact that this ‘unique’ probability measure is
unique only given an agent’s actual preferences between acts. Different agents
might well assign different probabilities to an event; indeed, quite arbitrary
assignments of probabilities are compatible with the Savage axioms.
With probability defined, all Savage needs to do is establish that VNM0–3
apply; this done he can apply the von Neumann-Morgenstern result to define
values and prove the EUT. This is straightforward: VNM0, VNM1 and VNM2
are easy consequences of S0, S1 and S3 respectively (though in general S6 is
needed to prove this) and the structure axiom VNM3 follows directly from S6.
S7 has a rather special place in Savage’s scheme: it is relevant only if we wish
to consider acts with countably infinitely many consequences (all the development so far, including the VNM axioms, has been in terms of finite-consequence
games). From S7 it can be proved that V is bounded, and as a consequence
of this that expected utility still represents preferences in the countably-infinite
case.
2.7
Objective chance
Technically speaking, the program outlined above seems in reasonably good
health: it is possible to quibble about the structure axioms or some of the principles of rationality, but in general the strategy of deriving probabilities from
preferences seems workable. However, it is less clear that the purely subjective notion of probability which emerges is a satisfactory analysis of probability.
In particular, it seems incompatible with the highly objective status played by
probability in science in general, and physics in particular. Whilst it is coherent
to advocate the abandonment of objective probabilities, it seems implausible:
it commits one to believing, for instance, that the predicted decay rate of radioisotopes is purely a matter of belief.
15
In the face of this problem, it has often been argued (Lewis 1980; Mellor
1971) that subjective probabilities can coexist with some other probabilities —
‘objective chances’, to use Lewis’s term. But how are these objective chances
to be understood? Papineau (1996) has identified two ways in which objective
chances are linked to non-probabilistic facts: an inferential link whereby we use
observed frequencies to estimate objective chances, and a decision-theoretical
link whereby we regard it as rational to base our choices on the objective
chances. In the presence of a subjective notion of probability (such as that
derivable from one of the decision-theoretic strategies above) we can formalise
this second link via Lewis’s Principal principle, which says — effectively — that
the rational subjective probability of a chance event occurring should be set
equal to its objective chance. From this principle we can derive the inferential
link also, at least in simple cases (see (Lewis 1980, pp. 106–108)).
But whether we have to justify Lewis’s Principle, or Papineau’s two links
directly, we owe some account of what sort of things objective chances are, and
why they should influence our behaviour the way that they do. The traditional
approach has been to identify objective chances with frequencies, but this generally leads to circularity (we wish it to be the case that it is very probable that
the frequency is close to the objective chance). Lewis has a more sophisticated
account (Lewis 1994) in which the objective chances are those given by the laws
which best fit the particular facts about the world (hence both frequency considerations, and those based on symmetry or simplicity, get to contribute), though
he acknowledges that he can see only “dimly” how these facts can constrain
rational action.
What is not acceptable (to Lewis, nor I believe to anyone) is simply to
introduce chance as a primitive concept and just stipulate that it is linked to
rational action in the required way. Lewis again:
Don’t call any alleged feature of reality “chance” unless you’ve already shown that you have something, knowledge of which could
constrain rational credence . . . I don’t begin to see, for instance, how
knowledge that two universals stand in a certain special relation N ∗
could constrain rational credence about the future coinstantiation
of those universals. Unless, of course, you can can convince me first
that this special relation is a chancemaking relation: that the fact
that N ∗ (J, K) makes it so, for instance, that each J has a 50%
chance of being K. But you can’t just tell me so. You have to show
me. Only if I already see — dimly will do! — how knowing the fact
that N ∗ (J, K) should make me believe to degree 50% that the next
J will be a K will I agree that the N ∗ relation deserves the name of
chancemaker that you have given it. (Lewis 1994, pp. 484–485)
If Lewis were wrong, of course — if it were legitimate just to posit that
certain quantities were objective chances — then there would be no probability
problem — and, in particular, as I will argue in section 3, there is no probability
problem in the Everett interpretation. Given the legitimacy (as argued for
by Saunders; see section 3.4) of regarding quantum branching as subjectively
16
indeterministic, we could just stipulate that some Quantum Principal Principle
held, linking rational belief ex hypothesi to mod-squared amplitude.
Papineau, in fact, turns this argument around: he argues that although it
is unsatisfactory merely to stipulate that mod-squared amplitude is objective
chance, no other account of objective chance does better! So it is changing the
goal-posts to claim that there is a probability problem in Everett but not in
classical physics: the two are on equal footing.
However, there is a more positive way to see the situation: if it could be
argued that quantum probabilities really should be treated as probabilities — if,
that is, some EUT could be derived for quantum mechanics, with mod-squared
amplitude in place of subjective probability — then not only would the problem
of probability in the Everett interpretation be resolved, but progress would have
been made on a wider philosophical problem with the very notion of objective
chance. This will be one goal of the remainder of the paper.
3
Applying decision theory to quantum events
To achieve this goal, we will need to transfer much of the machinery of classical
decision theory to a quantum context. Before this can be done, though, we need
to understand how a theory designed to deal with uncertainty between possible
outcomes deals with quantum events where all the outcomes are realised: this
will be the task of this section.
3.1
Quantum branching events
Our assumptions about the quantum universe can be summarised as follows:
• The Universe can be adequately modelled, at least for the purposes of
analysing rational decision-making, by assuming it to be a closed system,
described by a pure Hilbert-space state which evolves at all times according
to the unitary dynamics of quantum theory.
• In the Hilbert space of the theory, a de facto preferred basis is defined
by decoherence theory (see Zurek (1991) for introduction to this topic,
Zurek (2001) for a technical review, and Wallace (2001a, 2001b) for a
more philosophical analysis.) This basis is approximately diagonalized in
both position and momentum, and allows the Universe to be interpreted,
to a very good approximation, as a branching system of approximately
classical regions.
• Certain subsystems of the Universe can be interpreted as observers, who
exist within the approximately classical regions defined by the preferred
basis.
How are we to understand the branching events in such a theory? I have
argued elsewhere (Wallace 2001a) that they can be understood, literally, as
replacement of one classical world with several — so that in the Schrödinger
17
Cat experiment, for instance, after the splitting there is a part of the quantum
state which should be understood as describing a world in which the cat is alive,
and another which describes a world in which it is dead. This multiplication
comes about not as a consequence of adding extra, world-defining elements to
the quantum formalism, but as a consequence of an ontology of macroscopic
objects (suggested by Dennett 1991) according to which they are treated as
patterns in the underlying microphysics.
This account applies to human observers as much as to cats: such an observer, upon measuring an indeterminate event, branches into multiple observers
with each observer seeing a different outcome. Each future observer is (initially)
virtually a copy of the original observer, bearing just those causal and structural
relations to the original that future selves bear to past selves in a non-branching
theory. Since (arguably; see Parfit (1984) for an extended defence) the existence
of such relations is all that there is to personal identity, the post-branching observers can legitimately be understood as future selves of the original observer,
and he should care about them just as he would his unique future self in the
absence of branching.
3.2
The objective-deterministic viewpoint on quantum decisions
How should such an observer, if he is also a rational agent, choose between
various actions which cause branching? We will suppose for simplicity that the
agent has effectively perfect knowledge of the quantum state and the dynamics,
so that for each choice of action he might make, he can predict with certainty
the resultant post-branching quantum state. This might suggest that we need
only that part of decision theory dealing with decision-making in the absence
of uncertainty; call this the objective-deterministic (OD) viewpoint.
The decision theory available from the OD viewpoint is sparse indeed: all
we need is a transitive preference order over possible quantum states, then we
choose from the available acts that one which leads — with certainty, remember
— to the best available quantum state. If we adopt the Savage framework for
decision-making, for instance, the only axiom which survives is S1, which requires the preference ordering to be transitive: all the others deal with situations
involving uncertainty.
Clearly such a structure would not be rich enough to prove any interesting
results at all — let alone establish the quantum probability rule. However, even
if we cannot use the other Savage axioms we might hope to use close analogues of
them. Dominance, for instance (S4) suggests the following quantum analogue:
OD version of Dominance: Suppose an agent is about to be split into copies
C1 , . . . , CN , and then rewarded according to one of two possible rewardschemes P1 , P2 (so that either for each i the ith copy gets reward P1 (Ci )
for each i, or it gets P2 (Ci ).) If for no i will future copy Ci prefer P2 (Ci )
to P1 (Ci ), then the agent should not prefer P2 to P1 ; if in addition there
18
is at least one i for which P1 (Ci ) is preferred by copy Ci to P2 (Ci ), then
the agent should prefer P1 to P2 .
By no stretch of the imagination can this be construed as a statement of
classical decision theory — no such theory contains axioms dealing specifically
with agents who undergo fission! This is not to say that such a statement cannot
be defended as rational, but it would have to be treated as rational in the same
way that avoiding alligator pits is rational, as opposed to the way in which
avoiding intransitive preferences is rational.
In this case, a rational justification might go something like this:
I have the same structural and causal relation to my future copies
as I have to my future self in the absence of branching; hence insofar
as it is rational to care about my unique future self’s interest, it is
rational to care about the interests of my future copies. If I choose
P1 over P2 , no future copy will be worse off; hence P1 is at least as
good as P2 . If, further, at least one of my future copies is better off
under P1 than P2 , then I should choose P1 .
The problem with such justifications is that there are worryingly similar
justifications available for other assumptions which we must at all cost avoid
making. The obvious example would be the assumption that I should care
equally about all of my future selves irrespective of the amplitudes of their
respective branches (after all, they are all equally me, and none are cognizant of
the amplitudes of their, or other, branches). Obviously this assumption would
be disastrous for any recovery of the quantum rules, but it is arguably just as
intuitive as the quantum analogue of Dominance.9
3.3
Abandoning the OD viewpoint
The problem above is arguably not the worst for the OD viewpoint. There is
an important sense in which it asks the wrong question. It asks, in effect, how
we should act given the known truth of the Everett interpretation — akin to
asking how we should act if a matter-transporter were to be invented tomorrow.
But the real question should be: should we believe the Everett interpretation
in the first place?
In this case, it might be suggested that rationality considerations are just
irrelevant: if the Everett interpretation is the best available explanation of our
9 Anticipating the discussion of Section 8.1, I should note that there may be a more principled defence available for the application of the decision-theoretic axioms, even from the
perspective of the OD viewpoint: following a measurement the results of which I do not
know, my lack of knowledge of the result is just ordinary ignorance and I can apply decision
theory with impunity; then, invoking van Fraassen’s Reflection Principle, I can move those
probabilities backwards in time to the moment before the measurement. However, I do not
think this approach (which emerged in conversation with Hilary Greaves, to whom I am indebted here) can deal adequately with the issues to be raised in Sections 3.3 and 4.3; as such,
I shall not discuss it further.
19
empirical situation, adopt it; if not, don’t. This is too quick, though, as we can
see by analysing the notion of explanation in this context.
The sort of evidence that the Everett interpretation is being asked to explain
is along the lines of, “running such-and-such experiment N times led to M
clicks”; what sort of explanation could be given here? The simplest sort of
explanation to analyse would be one couched in terms of a deterministic (and
non-branching) theory and a known initial state: if the initial state is known
with certainty and it predicts the experimental results which in fact occur,
then (ceteris paribus, i. e. modulo considerations of simplicity and so forth) it
is explanatory of them. This is the sense in which Newtonian gravity explains
planetary motion, for instance.
But the prima facie randomness of quantum-mechanical empirical data suggests (again, if we ignore the possibility of branching) that either the initial
state, or the dynamical evolution, cannot be known with certainty: either we
have some probability distribution over initial states, or the dynamics is intrinsically indeterministic. In either case, the theory is explanatory if it predicts
that the experimental results are typical of the sort of results which would occur
with high probability.10
But if we wish to make a decision-theoretic analysis of the concept of probability, then statements about probability must be understandable in terms of
rational action (either directly, or through some Lewis-style Principal Principle).
The connection, qualitatively speaking, is that it is rational to act as though
events of very high probability will actually occur, and this suggests an account
of explanation directly in terms of rationality:
A theory T is explanatory of empirical data D (ceteris paribus)
if, had I believed T before collecting the data, it would have been
rational for me to expect D (or other data of which D is typical).
This account has the advantage of applying both to deterministic and indeterministic theories, and both to situations where we have perfect knowledge of
the initial state and situations where we do not. Can it be applied to the Everett interpretation, though? There seems no reason why not — provided that
we can make sense of the notion of rationality independently of the particular
theories under consideration. If we allow a theory to set its own standards of
rationality, then the account collapses into circularity: consider, for instance,
the theory that the world is going to end tomorrow and that it is rational to
believe that the world is going to end tomorrow.
This view of rationality does not commit us to the view that rationality itself
is not a legitimate subject of study and, where necessary, revision. Certainly
10 It would be conceptually simpler to regard a theory as explanatory, ceteris paribus, iff it
predicts that the experimental results themselves have high probability. But this is inadequate:
any given sequence of quantum events (or indeed coin-tosses) will usually have very low
probability. It is classes of such sequences (such as the class of all those with frequencies in a
given range) which are assigned high probability by explanatory theories. While the question
of how to analyse the notion of a “typical” result is rather subtle, the Everett interpretation
does not introduce any new problems to the analysis; I shall therefore not discuss it further.
20
we are physical systems, only imperfectly rational (Dennett 1987, pp. 94–99);
certainly we can consider and discuss what ideally rational behaviour is (as
shown by the lively debates over (for instance) the sure-thing principle and
Newcomb’s problem in the literature of decision theory). As we have learned
from Quine (1960), every part of our conceptual scheme is in principle open to
revision; however, this has to be done in a piecemeal way, keeping most of the
scheme fixed whilst certain parts are varied. (Recall Quine’s frequent reference
to Neurath’s metaphor of science as a boat, and his extension of the metaphor
to philosophy: we must rebuild the boat, but we have to remain afloat in it
whilst we do so; we can rebuild it plank by plank, but at any stage most of the
planks must be left alone.) In the case of theory change in physics, to vary both
our theory and the rational standards by which theories are judged would be to
go too far.
So, to analyse the Everett interpretation’s ability to recover the probability
rule — and so explain our empirical data — we need a viewpoint on rationality
which is not itself radically altered by the conceptual shift from single-world to
branching-world physics. To achieve this viewpoint we must abandon the OD
viewpoint and seek a new one.
3.4
The subjective-indeterministic viewpoint
To find our new perspective, we return to the situation of an agent undergoing
splitting. From our God’s-eye view we can regard the splitting as a deterministic
multiplication of the number of observers, but how should the agent himself view
it: if he is awaiting splitting, what should he expect ?
Saunders (1998) has argued persuasively that the agent should treat the
splitting as subjectively indeterministic: he should expect to become one future
copy or another but not both, and he should be uncertain as to which he will
become. His argument proceeds by analogy with classical splitting, such as that
which would result from a Star Trek matter transporter or an operation in which
my brain is split in two. It may be summarised as follows: in ordinary, nonbranching situations, the fact that I expect to become my future self supervenes
on the fact that my future self has the right causal and structural relations to my
current self so as to count as my future self. What, then, should I expect when
I have two or more such future selves? There are only three logical possibilities:
1. I should expect to become both future selves.
2. I should expect to become one or the other future self.
3. I should expect nothing: oblivion.
Of these, (3) seems absurd: the existence of either future self would guarantee
my future existence, so how can the existence of more such selves be treated
as death? (1) is at least coherent — we could imagine some telepathic link
between the two selves — but on any remotely materialist account of the mind
this link will have to supervene on some physical interaction between the two
21
copies which is not in fact present. This leaves (2) as the only option, and in
the absence of some strong criterion as to which copy to regard as “really” me,
I will have to treat it (subjectively) as indeterministic.
(In understanding Saunders’ argument, it is important to realise that there
are no further physical facts to discover about expectations which could decide
between (1-3): on the contrary, ex hypothesi all the physical facts are known.
Rather, we are regarding expectation as a higher-level concept supervenient
on the physical facts — closely related to our intuitive idea of the passage of
time — and asking how that concept applies to a novel but physically possible
situation).
Of course (argues Saunders) there is nothing particularly important about
the fact that the splitting is classical; hence the argument extends mutatis
mutandis to quantum branching, and implies that agents should treat their
own branching as a subjectively indeterministic event. We will call this the
subjective-indeterministic (or SI) viewpoint, in contrast with the OD viewpoint
which we have rejected for decision-theoretic purposes.11
But if branching is subjectively indeterministic, the agent can apply ordinary, classical decision theory without modification! The whole point of such
decision theory is to analyse decision-making under uncertainty, and from the SI
viewpoint — that is, from the viewpoint of the agent himself — that is exactly
the exercise in which he is involved when he is choosing between quantummechanical acts.
The SI viewpoint, then, is exactly what we need to judge the explanatory
adequacy of the Everett interpretation: it allows us to transfer the axioms
of decision theory directly across to the quantum-mechanical case. In the next
three sections we will show how this process is sufficient to establish the quantum
probability rule.
4
Quantum games and measurement neutrality
We have seen how decision theory offers two distinct routes to the ExpectedUtility representation theorem: through additivity of value (section 2.4), and
through a representation theorem for probabilities (section 2.6). We will shortly
see that both methods can be adapted straightforwardly to quantum mechanics
(with Deutsch’s proof effectively a form of the first method).
However, before this can be done we need to make sense of what, precisely,
are the quantum acts which we are considering. In this section, then, we will
define a certain large sub-class of quantum-mechanical acts, called “quantum
games”, and consider some properties of that class. This will be common ground
for the various derivations of expected utility presented in later sections, and
provides the framework whereby the subjective probabilities given by decision
11 This dichotomy of viewpoints — the “God’s-eye” and “personal” views — resembles that
presented by Sudbery (2000); his motivation and philosophy of mind, however, seem rather
different, as is his resolution of the probability problem.
22
theory are constrained to equal the probabilities predicted by quantum mechanics.
4.1
Quantum measurements
In the Everett framework, a measurement is simply one physical process amongst
many, and will be modelled as follows: let Hs be the Hilbert space of some subsystem of the Universe, and He be the Hilbert space of the measurement device;
b be a self-adjoint operator on Hs , with discrete spectrum.
let X
b consists of:
Then a non-branching measurement of X
1. Some state |M0 i of He , to be interpreted as its initial (pre-measurement)
state; this state must be an element of the preferred basis picked out by
decoherence.
b where X
b |λa i = xa |λa i . (Since we
2. Some basis |λa i of eigenstates of X,
allow for the possibility of degeneracy, we may have xa = xb even though
a 6= b.)
3. Some (orthogonal) set {|M; xa i} of “readout states” of Hs ⊗ He , also
elements of the decoherence basis, one for each state |λa i. The states must
physically display xa , in some way measurable by our observer (e. g. , by
the position of a needle).
4. Some dynamical process, triggered when the device is activated, and defined by the rule
|λa i⊗|M0 i −→ |M; xa i
(6)
(Since all dynamical processes in unitary quantum mechanics have to be
linear, this rule uniquely determines the process.)
What justifies calling this a ‘measurement’ ? The short answer is that it is the
standard definition; a more principled answer is that the point of a measurement
b is to find the value of X,
b and that whenever the value of X
b is definite,
of X
the measurement process will successfully return that value. (Of course, if the
b is not definite then the measurement process will lead to branching
value of X
of the device and the observer; but this is inevitable given linearity.)
The “non-branching” qualifier in the definition above refers to the assumption that the measurement device does not undergo quantum branching when
b is prepared in one of the states |λa i. This is a highly restrictive assumpH
tion, which we will need to lift. We do so in the next definition. a general
b consists of:
measurement of X
1. Some state |M0 i of He , to be interpreted as its initial (pre-measurement)
state; this state must be an element of the preferred basis picked out by
decoherence.
b where X
b |λa i = xa |λa i .
2. Some basis |λa i of eigenstates of X,
23
3. Some set {|M; xa ; αi} of “readout states” of Hs ⊗ He , also elements of
the decoherence basis, at least one for each state |λa i (the auxiliary label
α serves to distinguish states associated with the same |λa i). The states
must physically display xa , in some way measurable by our observer (e. g. ,
by the position of a needle).
4. Some dynamical process, triggered when the device is activated, and defined by the rule
X
|λa i⊗|M0 i −→
µ(λa ; α) |M; xa ; αi
(7)
α
where the µ(λa ; α) are complex numbers satisfying
P
α
|µ(λa ; α)|2 = 1.
In a general measurement, the measurement device (and thus the observer)
b has definite value; however, the observer can
undergoes branching even when X
predict that, in such a case, all his/her future copies will correctly learn the
b In practice, of course, most physical measurements are unlikely to
value of X.
be non-branching.
We end with three comments on the definition of measurement:
1. We are not restricting our attention to so-called “non-disturbing” measurements, in which |M; xa i = |λa i⊗|M′ ; xa i . In general measurements
will destroy or at least disrupt the system being measured, and we allow
for this possibility here.
2. In practice, the Hilbert space He would probably have to be expanded
to include an indefinitely large portion of the surrounding environment,
since the latter will inevitably become entangled with the device. We have
also not allowed for the possibility of the device having a large number of
possible initialisation states. But neither of these idealisations appear to
have any practical consequences for Deutsch’s argument, nor for the rest
of this paper.
3. Since a readout state’s labelling is a matter not only of physical facts
about that state but also of the labelling conventions used by the observer,
b and one of
there is no physical difference between a measurement of X
b where f is an arbitrary one-to-one function on the spectrum of
f (X),
b a measurement of f (X)
b may be interpreted simply as a measurement
X:
b
of X, using a different labelling convention. More accurately, there is a
physical difference, but it resides in the brain state of the observer (which
presumably encodes the labelling convention in some way) and not in the
measurement device.
To save on repetition, let us now define some general conventions for meab for the operator being measured, and denote
surement: we will generally use X
its eigenstates by |λa i; the eigenvalue of |λa i will be xa . (Recall that we allow
b so that may have xa = xb even though
for the possibility of degenerate X,
a 6= b.)
24
4.2
Defining quantum games
What form does the decision problem take for a quantum agent? Our (mildly
stylised) description of the problem in classical decision theory involved an agent
who was confronted with some chance setup and placed bets on the outcome.
This suggests an obvious quantum version: our agent measures some quantum
state, and receives a reward which depends on the result of the measurement.
For simplicity, let us suppose that the agent has perfect knowledge of the physical state that the Universe will be in, post-branching. Hence, he knows what
experiences all of his future copies will have, and what the amplitudes are for
each such experience. Nonetheless (as discussed in section 3.4) the process is
subjectively indeterministic for him: he expects to become one of the possible
future copies but does not know which.
Let us develop the details of this. Suppose that there exists some large class
of quantum systems, for which we will need to assume the following:
Q1 For each system in the class there exists at least one device capable of
measuring some discrete-spectrum observable of that system.
Q2 For each (positive integral) n, there exists some system with an n-dimensional
Hilbert space and some way of measuring a non-degenerate observable of
that system.
b on that
Q3 For any system in the class, and any measurable observable X
b
system, it is possible to perform any unitary transformation U which perb (that is, any U
b such that U
bX
bU
b † = f (X)
b ≡
mutes
the eigensubspaces of X
P
f
(x
)
|λ
i
hλ
|
for
some
real
function
f
).
a
a
a
a
b and Yb
Q4 On any pair of systems in the class, on which the operators X
are respectively measurable, it is possible to perform any joint unitary
b ⊗ Yb .
transformation which permutes the eigensubspaces of X
Q5 Any system in the class can be prepared in any pure state.
Thus for any system in the class, we can prepare it in an arbitrary state,
operate on it with a certain set of unitary operators, and measure it in some
basis. Operations of this form will be the chance setups of the quantum decision
problem, analogous to the set M of classical chance setups: we will denote the
set of such operations by MQ .
(Incidentally, the reader who finds the conditions Q1–Q5 too pedantic for
what is in any case supposed to be a set of operations which an agent can
contemplate performing and not necessarily a set of actually performable operations, is welcome to replace them with
Q∞ The class contains systems of every finite dimension, and any system in
the class can be prepared in any state and measured with respect to any
observable, and arbitrary unitary operations can be performed on systems
and pairs of systems in the class.)
25
Quantum acts (‘games’) then involve preparing a system, measuring it, and
then receiving some reward which is dependent on the result of the measurement.
It can be specified by the following rather cumbersome notation: an ordered
b P, ωi, where:
quartet h|ψi , X,
• |ψi is a pure state in the Hilbert space H of some system (for notational
simplicity the Hilbert space is not exhibited in the notation for a game);
b is a measureable observable on H with pure discrete spectrum;
• X
b to some set
• P, the payoff function, is a function from the spectrum of X
C of consequences;
• ω is the complete specification of a physical process by which
1. H is prepared in state |ψi;
b is made on H;
2. a measurement of X
3. In each branch where the measurement device shows xa , the consequence P(xa ) is given to the observer (in that branch).
We will suppose (the notion will be formalised later) that an agent’s preferences define a weak ordering on the space of games, so that an agent prefers to
play game G1 to game G2 iff G1 ≻ G2 . As usual, we will write G1 ≃ G2 just in
case neither game is preferred to the other. (Two such games will be referred to
as ‘value-equivalent’, pending the introduction of a quantitative value function.)
It will be completely crucial to later results that a game is merely specified
by such a quartet, not identified with it: games are certain sorts of physical
processes, not mathematical objects, and it is at this point open whether a
given game can be specified by more than one quartet. (This will in fact turn
out to be the case, with important consequences). Nonetheless, by construction
each quartet identifies a unique game, so we can consider preferences as holding
between quartets without ambiguity.
We will require that it is physically possible to realise games with arbitrary
payoff function P. This in turn requires the set of consequences to be specified
quite abstractly, with much of the physical details of how they are realised contained within the ω (this is, however, equally true for classical decision theory).
Note that whether or not a given physical process ω realises a given measureb
ment involves counterfactuals: to say that a device is a measuring device for X
is to say that it would fulfil our requirements for such a device, whatever state
was inserted into it. There is no counterfactual element to realising a payoff,
however: payoffs need be given only in those branches which are actually in
the superposition, so if with certainty eigenvalue x will not be recorded, then
the value of P(x) may be changed completely arbitrarily without affecting the
physical situation.
Deutsch’s own notation (among other simplifications) made no reference to
the physical contingencies specified in ω, tacitly assuming them to be irrelevant.
We can make this assumption explicit as:
26
Measurement neutrality: Rational agents are indifferent to the
physical details involved in specifying a quantum game; that is,
b P, ω1 i ≃ h|ψi , X,
b P, ω2 i for any ω1 , ω2 .
h|ψi , X,
If measurement neutrality holds, we can omit ω from the specification of a
game . Measurement neutrality is nowhere stated explicitly by Deutsch, but it
is tacit in his notation and central to his proof, as will be seen.
The set of quantum games, AQ , can now be defined:
AQ is the set of all physical processes labelled by ordered triples
b Pi where |ψi is a state of a preparable system, X
b is a meah|ψi , X,
surable observable of that system, and P is an arbitrary payoff funcb
tion on the spectrum of X.
In understanding this definition it is important to note that AQ is a set
of physical processes, not a set of ordered triples. Just as one triple may be
realised in many ways, so one and the same physical process may realise many
different triples. This fact is absolutely crucial to Deutsch’s proof, as section
4.4 will show.
4.3
Justifying measurement neutrality
At first sight there is scarcely any need to justify an assumption as obvious as
measurement neutrality: who cares exactly how a measurement device works,
provided that it works? What justifies this instinctive response is presumably
something like this: let A and B be possible measurement devices for some
b and for each eigenvalue xa of X
b let the agent be indifferent
observable X,
between the xa -readout states of A and those of B. Then if the agent is currently
planning to use device A, he can reason, “Suppose I get an arbitrary result xa .
Had I used device B I would still have got result xa , and would not care about the
difference caused in the readout state by changing devices; therefore, I should
be indifferent about swapping to device B.”
The only problem with this account is that it assumes that this sort of
counterfactual reasoning is legitimate in the face of (subjective) indeterminism,
and this is at best questionable (see, e. g. , Redhead (1987) for a discussion,
albeit not in the context of the Everett interpretation).
For a defence secure against this objection, consider how the traditional
Dirac-von Neumann description of quantum mechanics treats measurement. In
that account, a measurement device essentially does two things. When confronted with an eigenstate of the observable being measured, it reliably evolves
into a state which displays the associated eigenvalue. In addition, though, when
confronted with a superposition of eigenstates it causes wave-function collapse
onto one of the eigenstates (after which the device can be seen as reliably evolving into a readout state, as above).
In the Dirac-von Neumann description, it is rather mysterious why a measurement device induces collapse of the wave-function. One has the impression
that some mysterious power of the device, over and above its properties as a
27
reliable detector of eigenstates, induces the collapse, and hence it is prima facie possible that this power might affect the probabilities of collapse (and thus
that they might vary from device to device) — this would, of course, violate
measurement neutrality. That this is not the case, and that the probabilities
associated with the collapse are dependent only upon the state which collapses
(and indeed are equal to those stipulated by the Born rule) is true by fiat in the
Dirac-von Neumann description.
It is a strength of the Everett interpretation (at least as seen from the
subjective-indeterministic viewpoint) that it recovers the subjective validity of
the Dirac-von Neumann description: once decoherence (and thus branching)
occurs, subjectively there has been wave-function collapse. Furthermore there
is no “mysterious power” of the measurement device involved: measurement
devices by their nature amplify the superposition of eigenstates in the state to
be measured up to macroscopic levels, causing decoherence, and this in turn
leads to subjective collapse.
But this being the case, there is no rational justification for denying measurement neutrality. For the property of magnifying superpositions to macroscopic
scales is one which all measurement devices possess equally, by definition — so
if this is the only property of the devices relevant to collapse (after which the
system is subjectively deterministic, and so differences between measurement
devices are irrelevant) then no other properties can be relevant to a rational
allocation of probabilities. The only relevant properties must be the state being
measured, and the particular superposition which is magnified to macroscopic
scales — that is, the state being measured, and the observable being measured
on it.
4.4
Physical equivalence of different games
The following two axioms will be common to all the quantum decision theories
which we will present:
X0: Act availability The set of acts (i. e. , games) is AQ , as defined at the
end of section 4.2.
X1: Transitive preferences There exists a weak ordering ≻ on AQ which
satisfies measurement neutrality (that is, F ≃ G whenever acts F and G
b Pi).
are described by the same triple h|ψi , X,
Even with this minimal amount of decision theory, it is already possible to
derive one important set of results used in Deutsch’s proof. These results involve
the realisation that certain quantum games, described by different labels in our
b and P) are in reality the same
notation (that is, with different choices of |ψi, X
game.
The essential idea used in all of these proofs is this: measurement neutrality
asserts that the correct description of a game qua game is given by the state,
the measured operator and the payoff, and that the remaining physical details
are irrelevant. But this is not the right way to carve up the space of games
28
qua physical systems: one game can be realised in many ways, but also one
and the same physical process may be understood as realising many different
games. By going back and forth between the two sorts of indifference implied
by measurement neutrality and by existence of multiple labels for the same
game (implied by the physics of playing the game), we are able to prove our
equivalences.
We begin with the following result:
Payoff Equivalence Theorem (PET): Let P be a payoff scheme
b and let f be any function from the spectrum of X
b to the reals
for X,
satisfying
f (xa ) = f (xb ) −→ P(xa ) = P(xb ).
(8)
(Hence the function P · f −1 is well-defined even though f may be
b Pi and h|ψi , f (X),
b P ·f −1 i
non-invertible.) Then the games h|ψi , X,
can be realised by the same physical process; they therefore have the
same value.
Proof: Recall that our definition of a measurement process involves a set of
states |M; xa i of the decoherence-preferred basis, which are understood as readout states — and that the rule associating an eigenvalue xa with a readout state
|M; xa i is just a matter of convention. Change this convention, then: regard
|M; xa i as displaying f (xa ) — but also change the payoff scheme: replace a payoff P(f (xa )) upon getting result f (xa ) with a payoff (P · f −1 )(f (xa )) ≡ P(xa ).
b Pi with h|ψi , f (X),
b P · f −1 i with
These two changes replace the game h|ψi , X,
f satisfying (8) — but no physical change at all has occurred, just a change of
labelling convention.
b Pi — it
But the value function does not assign values to triples h|ψi , X,
assigns them to acts, construed as physical sequences of events. If two “different”
games can be realised by the same sequence of events, then, they are really the
same game, and should be assigned the same value. Measurement neutrality
then tells us that all other realisations of the same game — that is, all other
b P, ωi and h|ψi , f (X),
b P · f −1 , ω ′ i — are also value-equivalent.
quartets h|ψi , X,
2
A similar physical equivalence holds between transformations of the state
and of the operator to be measured.
Measurement Equivalence Theorem (MET):
b be any unitary operator which permutes, possibly triv1. Let U
b i. e. X
bU
b |λa i = π(xa )U
b |λa i,
ially, the eigensubspaces of X:
b Then the
where π is some permutation of the spectrum of X.
†
b
b
b
b
b
b Pi
games h|ψi , X, Pi and hU |ψi , U X U , Pi ≡ h|ψi , π −1 (X),
have the same value.
b is nondegenerate and let f be a permu2. In particular, suppose X
b f by U
b f |λa i =
tation of its spectrum: f (xa ) ≡ xπ(a) . Define U
29
λπ(a) . Then the games h|ψi , X,
b Pi and hU
b f |ψi , f −1 (X),
b Pi
have the same value.
Proof: (2) is an immediate corollary of (1), which we prove as follows. (Note
that the realisability of the unitary transformations in (1) and (2) follows from
b be x1 , . . . xM ; let d(i) be
assumption Q3.) Let the distinct eigenvalues of X
b We change our labelling for eigenthe dimension of the xi -eigensubspace of X.
vectors, denoting them by |xi ; ji, where j is a label ranging from 1 to d(xi ).12
b carries this basis to another eigenbasis of X,
b whose elements are similarly
U
b
b forces
denoted |∗; xi ; ji: U |xi ; ji = |∗; π(xi ); ji. (Note that the unitarity of U
d(xi ) = d(π(xi )), so this is well-defined.)
b |ψi , U
bX
bU
b † , Pi by the following process:
Let us realise the game hU
1 Prepare the system in state |ψi, so that the overall quantum state is


Xi )
X d(x
αi,j |xi ; ji ⊗ |M0 i
(9)
|ψi ⊗ |M0 i ≡ 
j=1
i
where |M0 i is the initial state of the measurement device.
b , changing the
2a Operate on the state to be measured with the operator U
overall state into state


Xi )
X d(x

αi,j |∗; π(xi ); ji ⊗ |M0 i .
(10)
i
j=1
b using the following dynamics:
2b Measure π −1 (X)
|∗; xi ; ji⊗|M0 i −→ M; π −1 (xi ); j
(11)
where for each xi , the states |M; xi ; ji are a set of readout states giving
readout xi . (This fits our definition of a non-branching measurement
process; the generalisation to a branching process is trivial.)
3 The final state is now
Xi )
X d(x
i
j=1
αi,j |M; xi ; ji .
(12)
In the branches in which result xi is recorded, give a reward of value P(xi ).
12 We
implicitly restrict to finite-dimensional Hilbert spaces, but the generalisation is trivial
b has pure discrete spectrum: just replace the finite set of indices used to label
provided that X
degenerate eigenvectors with an infinite set.
30
b |ψi is
This is indeed a realisation of the game: in steps 1 and 2a the state U
b is measured, and in step 3 the payoff
prepared, in step 2b the operator π −1 (X)
is made. But now suppose that we avert our eyes from the dynamical details
of steps 2a and 2b, considering them to be a black box. Then the effect of this
black box is simply to carry out the transformation


Xi )
X d(x
Xi )
X d(x


αi,j |M; xi ; ji .
(13)
αi,j |xi ; ji ⊗ |M0 i −→
i
i
j=1
j=1
b and we
This transformation, though, fits the definition for a measurement of X,
have an alternative description of the same physical events: in step 1 the state
b is measured, and in step 3
|ψi is prepared, in steps 2a and 2b the operator X
b Pi — so again we
payoff is made. This is a realisation of the game h|ψi , X,
have two different games realised by the same physical process and thus being
assigned the same value. 2
Two important (and immediate) corollaries of the MET concern the role of
symmetry.
b is defined as
Operator Symmetry Principle: Suppose that U
†
bX
bU
b =X
b (equivalently, supfor part (1) of the MET, and that U
b |ψi , X,
b Pi and
pose that the permutation π is trivial). Then hU
b
h|ψi , X, Pi have the same value.
b is non-degenerate,
State Symmetry Principle: Suppose that X
b
that U f is defined as for part (2) of the MET, and that |ψi is inb f ; that is, that U
b f |ψi = |ψi. Then
variant under the action of U
b Pi and h|ψi , f (X),
b Pi have the same value.
h|ψi , X,
In other words, the symmetries of a state being measured imply relationships
between the values of measuring different observables upon that state, and vice
versa.
The next equivalence theorem we prove is a corollary of the PET.
b and X
b ′ have the
Operator equivalence theorem (OET): Let X
same spectrum,and suppose that they have a certain set of eigenb and X
b ′ agree on the subspace S spanned
states in common. Let X
by those eigenstates, and let |ψi ∈ S. Define P and P ′ to be payoff
b and X
b ′ respectively, which agree on
functions on the spectra of X
b S.
the spectrum of X|
b Pi ≃ h|ψi , X
b ′ , P ′ i.
Then h|ψi , X,
Proof: Without loss of generality, assume that 0 is in the spectrum neither of
b nor X
b ′ . We define:
X
31
b 0 is the operator equal to X
b on S and equal to zero otherwise (clearly
• X
′
b
b
X 0 = X 0 ).
b defined by f (x) = x for x in the
• f is that function on the spectrum of X
b 0 , and f (x) = 0 otherwise.
spectrum of X
• N is some arbitrary consequence.
b such that P1 (x) = P(x) whenever x is in
• P1 is a payoff function for X,
b
the spectrum of X|S and P1 (x) = N otherwise.
b 0 , such that P0 (x)=P1 (x) for x 6= 0 and
• P0 is a payoff function for X
P0 (0) = N .
As was explained after the definition of a game, payoffs are not specified
counterfactually, and hence the value of a payoff function is arbitrary on any
xa which with certainty will not occur. Hence without changing the game as a
physical process, we can replace the payoff function P by P1 (since the observer
b S ).
will, with certainty, get one of the results in the spectrum of X|
Now we apply the PET:
b P
b 1 ) = V(|ψi , f (X),
b P1 · f −1 ).
V(|ψi , X,
(14)
b P
b i ≃ h|ψi , X
b 0, P
b 0 i.
h|ψi , X,
(15)
b =X
b 0 and P0 = P1 · f −1 , so in fact we have
But f (X)
An identical argument tells us that
′
′
b ,P
b i ≃ h|ψi , X
b 0, P
b 0 i,
h|ψi , X
(16)
and the theorem follows. 2
Something should be said about the role of the OET in Deutsch’s proof. It
appears to be necessary to the proof unless we are to make quite strong spectral
b but is not explicitly used. It might be that Deutsch
assumptions about X,
avoids using either by defining measurements (tacitly) in a state-dependent
b on the state P αa |λa i could have been defined as
way: a measurement of X
a
any transformation with final state of form
X
αa |M; xa i .
(17)
a
However, the notion of measurement defined in section 4.1 was intentionally
state-independent (and thus counterfactual): that is, whether something does
or does not count as a measurement device does not depend on which microstate
triggers it. This seems intuitively reasonable: after all, a device which always
emits the result “spin up” would not count as a legitimate spin-measurement
device even if the state it was measuring happened to have spin up!
The next equivalence theorem we prove shows that it is essentially the amplitudes of results and not the details of the state which matter.
32
b and X
b ′ be self-adjoint
State equivalence theorem (OET): Let X
b |λa i = xa |λa i and
operators with discrete spectrum such that X
P
PN
′
N
′
′
b |λ′a i = xa |λ′a i; let |ψi =
X
a=1 |λa i and |ψ i =
a=1 |λa i; Let P
′
′
b and X
b respectively, which agree on the set
and P be payoffs for X
{x1 , . . . , xN }.
b Pi ≃ h|ψ ′ i , X
b ′ , P ′ i.
Then h|ψi , X,
Note that |ψi and |ψ ′ i need not be in the same Hilbert space, and that
b
|λ1 i , . . . , |λN i (|λ′1 i , . . . , |λ′N i) may not be the full set of eigenvectors for X
′
b ).
(X
Proof: Assume without loss of generality that Dim(H) ≤ Dim(H′ ). From the
b with an operator Yb , which agrees with X
b on the span
OET, we may replace X,
of the |λa i and which is non-degenerate elsewhere, with eigenvectors |µi i and
distinct eigenvalues yi , with yi 6= xa for all i, a: thus we have
X
X
yi |µi i hµi | .
(18)
Yb =
xa |λa i hλa | +
a
i
′
b with
Similarly, we may replace X
X
X
X
′
Yb =
xa |λ′a i hλ′a | +
yi |µ′i i hµ′i | +
zj |νj i hνj | .
a
i
(19)
j
′
(The third sum in the definition of Yb occurs because the dimension of H′ may
exceed the dimension of H; for convenience we will require that the zj are all
distinct from one another and from the xa and yi .)
Let P1 be a payout scheme which coincides with P (and thus P ′ on the set
′
{x1 , . . . , xN }). Let P1′ be a payout scheme for Yb which coincides with P1 on
the spectrum of Yb .
Now consider the following operation (which is performable given Q4 and
Q5):
1. Prepare H in an arbitrary state |ψi, and H′ in some fixed state |0′ i.
2. Perform a joint operation on H ⊗ H′ , defined by:
|λa i⊗|0′ i −→ |0i⊗|λ′a i ;
b |µi i⊗|0′ i −→ |0i⊗|µ′i i
U
where |0i is some fixed state of H. At the end of this process, the joint
state is |0i⊗|ψ ′ i for some state |ψ ′ i; discard the fixed state |0i. (If H = H′ ,
replace this operation with the simpler one |λa i → |λ′a i, |µi i → |µ′i i, which
just leaves H in state |ψ ′ i.)
33
′
3. Measure Yb by some process
|λ′a i⊗|M0 i −→ |M; xa i ;
|µ′i i⊗|M0 i −→ |M; yi i ;
′
νj ⊗|M0 i −→ |M; zj i .
4. In the branch where the result is x, provide a payout P1′ (x).
As usual in these proofs, this scheme admits of two descriptions. If we regard
step 1 as the preparation of state |ψi ∈ H and steps 2–3 as a measurement of
Yb for that state, the process instantiates the game h|ψi , Yb , P1 i. If however we
regard 1–2 as the preparation of the state |ψ ′ i ∈ H′ , and 3 as a measurement of
′
′
Yb on that state, then the process instantiates h|ψ ′ i , Yb , P1′ i. These games are
thus of equal value; hence when |ψi is as in the statement of the SET (so that
b Pi and h|ψ ′ i , X
b ′ , P ′ i. 2
we can apply the OET) so are h|ψi , X,
4.5
The Grand Equivalence Theorem
The results of the previous section, jointly, have powerful consequences for
decision-making, and all are utilized (tacitly) at various points in Deutsch’s
proof. However, it is possible to take them further than Deutsch does: together,
they imply a very powerful result of which they in turn are immediate consequences. To state this result, recall that the weight of a branch is simply the
squared modulus of the amplitude of that branch (relative to the pre-branching
amplitude, of course); thus if the state of a measuring device following measurement is
X
αa |M; xa i ,
(20)
a
then the weight of the branch in which result xa occurs is |α|2 . In a game, the
weight of a consequence will be defined as the sum of the weights of all branches
in which that consequence occurs.
Grand Equivalence Theorem For decision purposes, a game is
completely specified by giving all the distinct possible consequences
of that game, together with their weights.
b Pi has N possible consequences c1 , . . . , cN
Proof: Suppose a game h|ψi , X,
with weights w1 , . . . wN , then we will show that it is equivalent to what I will
b 0 , P0 i, where
call a ‘canonical’ game, h|ψ0 i , X
• |ψ0 i is a state in an N -dimensional Hilbert space H0 . (That such games
exist follows from Q2).
b 0 = PN n |ni hn| .
• X
n=1
PN √
• |ψ0 i = n=1 wn |ni .
34
• P0 (n) = cn .
If each game with the same set of consequences and weights is equivalent to
the same canonical game, then they are all equivalent to each other and the
theorem will follow.
We proceed as follows. For each cn , let Mn be the set of all eigenvalues of
b for which P(x) = cn , and let M0 be the set of all eigenvalues which (given
X
|ψi) will not be found to occur in any branch post-measurement. Since payoff
functions are not specified counterfactually, we can replace P by any payoff
function P ′ which is constant (say, equals c0 ) on M0 .
b with eigenvalues
Let Sn be the subspace spanned by all eigenvectors of X
in Mn , and S0 the subspace spanned by eigenvectors with eigenvalues in M0 .
b by f (x) = j whenever x ∈ Mj . By
Define the function f on the spectrum of X
b with
the PET, we can replace X
b′ =
X
N
X
jPj ,
(21)
j=0
where Pj is the projector onto Sj , and P ′ by P ′′ , where P ′′ (j) = cj .
b with eigenvalue
For each n ≥ 1, let |ψn i be an arbitrary eigenstate of X
√
in Mn . The vector Pj |ψi has amplitude wj (up to phase), and any unitary
b fixed. Thus by the
transformation which leaves each Sj fixed will also leave X
Operator Symmetry Principle we can replace |ψi by
|ψ ′ i =
N
X
√
wn |ψn i .
(22)
n=1
b ′ , P ′′ i. We now
Thus we have shown that our game is equivalent to h|ψ ′ i , X
apply the SET to conclude that this is in turn equivalent to our canonical game.
4.6
Composite games
At various points in Deutsch’s paper, it is necessary to make use of the idea of
‘composite’ games: that is, games where a quantum state is measured by where,
instead of giving a payoff dependent on the result of the measurement, another
game is played, where the game is dependent on the result of the measurement:
effectively, the payoff function P takes values not in C but in AQ . Composite
games, of course, can themselves be composed, and so forth: the set of all such
composite games will be denoted by ACQ .
However, given measurement neutrality the decision problem is not really
extended by the move from AQ to ACQ . For any composite game can be
replaced by one huge simple game, with all the states which might be needed in
sub-games prepared in advance and one (very complex) measurement performed
on all of them at once. It then follows from measurement neutrality (most
directly from the GET) that this replacement does not change the value of the
game.
35
5
Deutsch’s Proof
Deutsch’s proof of the expected-utility rule is best understood as a “quantization” of the additivity proof of the EUT given in section 2.4: it makes essential
use of the additivity of consequences. (Deutsch’s own discussion of additivity
(Deutsch 1999, p. 5) seems to imply that additivity is a mere convention, but the
discussion of section 2.4 should make it clear that additivity places strong constraints on preferences.) In this section I will give a reconstruction of Deutsch’s
own proof; I will then reformulate Deutsch’s axioms slightly in order to present
an alternative and more direct additivity proof. I conclude the section with a
brief discussion of how Deutsch’s proof fails in the presence of ‘hidden variables’.
5.1
Deutsch’s postulates
Deutsch develops his theory entirely within the framework of an additive value
function on consequences; he does not, however, explicitly make the assumption
that the value of sequential acts is additive. In fact, in his system the values
of acts are derivable from the values of the consequences, given a qualitative
preference order on acts: any constant act (where the act, with certainty, leads
to some consequence c) is assigned value V(c),13 and another act A is assigned
value V(c) iff the agent is indifferent between performing A and performing the
constant act. This requires there to exist quite a large class of consequences,
one for each real number between the largest- and smallest-value consequence.
Deutsch’s class of acts is the composite set ACQ ; in his system, the value
of composite games follows from the value of simple ones. This follows from
b Pi, and if G ′
his Substitutivity assumption (p. 5 of his paper): if G = h|ψi , X,
is the composite game played by measuring |ψi and then playing game G(x) if
we get result x, then a sufficient (but not necessary) condition for G ≃ G ′ is
b In words, this means
that V[P(xa )] = V[G(xa )] for all xa in the spectrum of X.
that an agent is indifferent between a game where on some outcome he receives
reward c, and another where on that same outcome he plays a game which is
worth c. Fairly clearly, this is another form of the Sure-Thing Principle.
Substitutivity allows us to simplify the notation for games: let P, P ′ be any
b such that P(xa ) ≃ P ′ (xa ). Then substitutivity
payoff functions for the same X
tells us that irrespective of the state,
b Pi ≃ h|ψi , X,
b P ′ i.
h|ψi , X,
(23)
This means we can ignore the details of what the consequences are, and just use
their numerical values: henceforth, then, P will be taken to be a function from
b into the reals.
the spectrum of X
The main use which Deutsch makes of Additivity is to prove the
13 Technically, we should prove that this assignation is consistent, given that many different
acts might all have constant consequences: this is a trivial consequence of MET and PET,
though.
36
Additivity Lemma:
b P + k) = V(|ψi , X,
b P) + k.
V(|ψi , X,
(24)
He does so by considering the physical process of playing some quantum game
b Pi and then receiving a reward of value k with certainty. This
G = h|ψi , X,
is physically identical to measuring |ψi and then receiving two rewards upon
getting result xa : one of value P(xa ) and one of value k; by additivity this is
equivalent to receiving a single reward of value P(xa ) + k.
To complete this proof Deutsch needs a second use of additivity, to show
that the value of playing G and then receiving k must be V(G) + k; however,
this requires him to assume value to be additive across acts and not just consequences. This would be a fairly innocuous extra assumption for him to make;
for completeness, though, note that his result can be also be derived from additivity of consequences and substitutivity: assume that the fixed reward k is
received before playing G, then note by substitutivity that receiving k and then
playing G must have the same value as receiving k and then receiving a reward
worth V(G), and that by additivity this latter process is worth k + V(G). The
result then follows.
Deutsch requires two further assumptions. One is a dominance principle
(tacitly introduced on page 12): if P(xa ) ≥ P ′ (xa ) for all xa , then a game
using P as payoff scheme is preferred or equivalent to one using P ′ . (A similar
assumption, recall, was used in the classical discussion of additive value; note
that the equivalence of two games with equal-valued consequences for each given
state, and hence the replacement of P with a real-valued function, follows as
easily from Dominance as from Substitutivity.)
The last of Deutsch’s assumptions, the Zero-Sum Rule, is on the face of
it more contentious. This rule states that for any game G, if the payoff P is
replaced by −P then the value of the game is replaced by −V(G). This is a trivial
consequence of consequence additivity for constant games but it is unclear what
Deutsch’s motivation for it is when considering general games; in section 5.3 we
shall derive it from act additivity. Saunders has argued for the Zero-Sum Rule
directly by considering the situation from the viewpoint of the banker:
. . . banking too is a form of gambling. The only difference between
acting as one who bets, and as banker who accepts the bet, is that
whereas the gambler pays a stake in order to play, and receives
payoffs according to the outcomes, the banker receives the stake in
order to act as banker, and pays out the payoffs according to the
outcomes. The zero-sum rule is the statement that the most one
will pay in the hope of gaining a utility is the least that one will
accept for fear of losing it. (Saunders 2002)
Deutsch’s postulates can then be axiomatized as follows.
D0: Act availability The set of acts is ACQ .
37
D1: Transitive preferences (acts) There is a weak order ≻ on ACQ which
satisfies measurement neutrality (and which defines a weak order on C via
the constant acts).
D2: Additive preferences (consequences) There is a composition operation + on C such that ≻ is additive with respect to + (that is, satisfies
A1–A5). (Hence there exists some value function V on consequences.)
D3: Consequence availability Each act in ACQ is value-equivalent to some
constant act. (Hence V can be extended to acts.)
D4: Substitutivity Forming a compound game from any game, by substituting for its consequences games of equal value to those consequences, does
not change the value of that game.
b P) = −V(|ψi , X,
b −P).
D5: Zero-sum Rule V(|ψi , X,
D6: Dominance If P ≥ P ′ then
5.2
b P) ≥ V(|ψi , X,
b P ′ ).
V(|ψi , X,
Deutsch’s Proof
In this section, we put together the results so far to achieve Deutsch’s goal: to
prove the quantum probability rule. The proof given below follows Deutsch’s
own proof rather closely, although some minor changes have been made for clarity or to conform to my notation and terminology. As such, it uses the various
equivalence theorems of section 4.4, rather than the single Grand Equivalence
Theorem.
b with some eigenvalue
As usual, |λa i will always denote an eigenstate of X
b
xa . By default, the operator measured will be X and the payoff function will
b thus, h|ψii ≡ h|ψi , X,
b Pi.
be f (x) = x, restricted to the spectrum of X;
√
Stage 1 Let |ψi = (1/ 2)(|λ1 i + |λ2 i). Then V(|ψi) = 1/2(x1 + x2 ).
We know (from the Additivity Lemma) that Deutsch can show that
b P + k) = V(|ψi , X,
b P) + k.
V(|ψi , X,
(25)
b + k) = V(|ψi , X)
b + k.
V(|ψi , X
(26)
b = −V(|ψi , X),
b
V(|ψi , −X)
(27)
b + k) = −V(|ψi , X)
b + k.
V(|ψi , −X
(28)
The PET simplifies this to
Similarly, the Zero-Sum Rule together with another use of the PET gives us
and combining these gives
38
Now, let f be the function of reflection about the point 1/2(x1 + x2 ). Then
b is non-degenerate and that the spectrum
f (x) = −x + x1 + x2 . Provided that X
b
b f as in
of X is invariant under the action of f , we can define the operator U
section 4.4. Since |ψi is a symmetry of |ψi, the State Symmetry Principle tells
us that
b + x1 + x2 ) = V(|ψi , X).
b
V(|ψi , −X
(29)
Combining this with (28), we have
b = −V(|ψi , X)
b + x1 + x2 ,
V(|ψi , X)
(30)
b = 1/2(x1 + x2 ), as required.
which solves to give V(|ψi , X)
b is degenerate, or has a spectrum which is not
In the general case where X
b with X
b ′ , which
invariant under the action of f , we use the OET to replace X
b on the span of {|λ1 i , |λ2 i} and equals 1/2(x1 + x2 ) times the
agrees with X
identity otherwise. The result then follows, except in the case where x1 = x2 ;
in this case the result is trivial.
Deutsch refers to this result, with some justice, as ‘pivotal’: it is the first
point in the proof where a value has been calculated for a superposition of
different-value states, and the first time in our discussion of decision theory
that we have forced the probabilities to take specific values, independent of the
subjective views of our agent.
It is crucial to understand the importance in the proof of the symmetry of
|ψi under reflection, which in turn depends on the equality of the amplitudes in
the superposition; the proof would fail for |ψi = α |λ1 i + β |λ2 i , unless α = β.
√
Stage 2 If N = 2n for some positive integer n, and if |ψi = (1/ N )(|λ1 i +
· · · + |λN i), then
V(|ψi) = (1/N )(x1 + · · · + xN ).
(31)
The proof is recursive on n, and I will give only the first step (the generalisation is obvious). Define:
• |ψi = (1/2)(|λ1 i + |λ2 i + |λ3 i + |λ4 i)
√
√
• |Ai = (1/ N )(|λ1 i + |λ2 i); |Bi = (1/ N )(|λ3 i + |λ4 i)
• yA = (1/2)(x1 + x2 ); yB = (1/2)(x3 + x4 ).
• Yb = yA |Ai hA| + yB |Bi hB| .
Now, the game G = h|ψi , Yb i has value 1/4(x1 + x2 + x3 + x4 ), by Stage
1. In the yA branch, a reward of value 1/2(x1 + x2 ) is given; by Substitutivity
the observer is indifferent between receiving that reward and playing the game
b since the latter game has the same value. A similar observation
GA = h|ψi , Xi,
applies in the yB branch.
39
So the value to the observer of measuring Yb on |ψi and then playing either
GA or GB according to the result of the measurement is 1/4(x1 + x2 + x3 + x4 ).
But the physical process which instantiates this sequence of games is just
!
4
4
X
X
1
1
|λi i ⊗ |M0 i →
|M; xi i ,
(32)
2
2
i=1
i=1
b hence, the result follows.
which is also an instantiation of the game h|ψi , Xi;
Stage 3 Let N = 2n as before, and let a1 , a2 be positive integers such that
√
√
a1 + a2 = N . Define |ψi by |ψi = √1N ( a1 |λ1 i + a2 |λ2 i). Then
V(|ψi) =
1
(a1 x1 + a2 x2 ).
N
(33)
b on a general superposition α |λ1 i + β |λ2 i of |λ1 i and
One way to measure X
|λ2 i is to use an N -dimensional auxiliary Hilbert space Ha spanned by states
|µi i, and prepare it in state
a1
N
X
1
1 X
|µi i
|µi i or |2i = √
|1i = √
a1 i=1
a2 i=a +1
(34)
1
b takes value x1 or x2 ; this process can be combined
according to whether X
with the erasure and destruction
of the initial state. If we then define a nonP
degenerate operator Yb = i yi |µi i hµi | on HA and measure it, the overall dynamical process will be
a1
α X
β
α |λ1 i + β |λ2 i −→ α |1i+ β |2i −→ √
|M; yi i+ √
N i=1
N
N
X
i=a1 +1
|M; yi i . (35)
If a payoff of x1 is provided whenever the measurement readout is yi for i ≤ a1 ,
b
and x2 otherwise, then the overall process realises the game hα |λ1 i + β |λ2 i , Xi.
However, the selfsame process can also be regarded as a realisation of the
measurement of the degenerate observable f (Yb ) (where f (yi ) = x1 for i ≤ a1
and f (yi ) = x2 otherwise) on the state
|φi = α |1i + β |2i .
(36)
p
p
In the particular case where α = a1 /N and β = a2 /N , then |φi = (1/N )(|µ1 i+
· · ·+|µN i, and this second game has value (1/N )(a1 x1 +a2 x2 ), by Stage 2; hence
the result is proved.
Deutsch then goes on to prove the result for arbitrary N (i. e. , not just
N = 2n ); however, that step can be skipped from the proof without consequence.
√
Stage
4 Let a be a positive real number less than 1, and let |ψi = a |λ1 i +
√
1 − a |λ2 i. Then V(|ψi) = ax1 + (1 − a)x2 .
40
Suppose, without loss of generality, that x1 ≤ x2 , and make the following
definitions:
• G = h|ψii.
• {an } is a decreasing sequence of numbers of form an = An /2n , where An
is an integer, and such that limn→∞ an = a. (This will always be possible,
as numbers of this form are dense in the reals.)
√
√
• |ψn i = an |λ1 i + 1 − an |λ2 i.
√
√
√
• |φn i = (1/ an )( a |λ1 i + an − a |λ2 i .
• Gn = h|ψn ii.
• Gn′ = h|φn ii.
Now, from Stage 3 we know that V(Gn ) = an x1 + (1 − an )x2 . We don’t know
the value of Gn′ , but by the postulate of Dominance we know that it is at least
x1 . Then, by Substitutivity, the value to the observer of measuring |ψn i, then
receiving x2 euros if the result is x2 and playing Gn′ if the result is x1 , is at least
as great as the V(Gn ).
But this sequence of games is, by strong measurement neutrality, just a
realisation of G, for its end state is one in which a reward of√x1 euros is given
with amplitude a and a reward of x2 euros with amplitude 1 − a. It follows
that V(G) ≥ V(Gn ) for all n, and hence that V(G) ≥ ax1 + (1 − a)x2 .
A similar argument with an increasing sequence establishes that V(G) ≤
ax1 + (1 − a)x2 , and the result is proved.
Stage 5 Let α1 , α2 be complex numbers such that |α1 |2 + |α2 |2 = 1, and let
|ψi = α1 |λ1 i + α2 |λ2 i. Then V(|ψi) = |α1 |2 x1 + |α2 |2 x2 .
This is an immediate
consequence of the Operator Symmetry Principle, as
b invariant.
b = P exp(iθa ) |λa i hλa | leaves X
the operator U
a
Stage 6 The quantum probability
rule is the correct
P strategy to determine prefP
erence: that is, if |ψi = i αi |λi i, then V(|ψi) = i |αi |2 xi .
This last stage of the proof is simple and will not be spelled out in detail. It
proceeds in exactly the same way as the proof of Stage 2: any n-term measurement can be assembled by successive 2-term measurements, using Substitutivity
and weak measurement neutrality.
5.3
Alternate form of Deutsch’s proof
In this section I shall give an alternative proof of Deutsch’s result. It differs
from Deutsch’s own proof in two ways: the Grand Equivalence Theorem is used
directly, and additivity of consequences is replaced by additivity of acts. The
latter is a mild strengthening of Deutsch’s axioms, but seems fairly innocuous
41
— and, in any case, seems substantially more plausible than the Zero-sum rule
even though strictly it implies it. It allows us to streamline the axiomatization, removing reference to compound games and making the axioms virtually
identical to the classical structure U0–U4.
In the classical case, we used act additivity to combine bets to construct an
arbitrary bet. We can do so in the quantum case also, if we allow that games
may include not just the holistic process of preparing, betting on and measuring
a quantum state, but also that of betting on a quantum state which has already
been prepared, and which is to be measured by another party. If so, then act
additivity implies that the value of placing such a bet is unaffected by which
bets have already been placed: this means that
b P1 ) + V(|ψi , X,
b P2 ) = V(|ψi , X,
b P1 + P2 ).
V(|ψi , X,
(37)
Our new axiom scheme, then, will be:
D0′ : Act availability The set of acts is AQ .
D1′ : Transitive preferences There is a weak order ≻ on AQ which satisfies
measurement neutrality (and which defines a weak order on C via the
constant acts).
D2′ : Dominance If P(xa ) P ′ (xa ) for all xa then
b Pi h|ψi , X,
b P ′ i.
h|ψi , X,
(38)
(M, P1 ) + (M, P2 ) = (M, P1 + P2 ).
(39)
D3′ : Composition There is an operation + of composition on C, and another
such operation + on AQ such that
D4′ : Act additivity The weak ordering ≻ on AQ is additive (that is, satisfies
A1–A5) with respect to composition.
This list is very similar to U0–U4: the only real differences are the use of
the quantum acts AQ and the assumption of measurement neutrality.
Our new proof is as follows. U4 implies the existence of an additive value
function V on acts, and hence (via constant
Pacts) on consequences. We define
the expected utility of a game by EU (G) = i wi Vi , where the sum ranges over
the distinct numerical values of the consequences with non-zero weight (i. e. the
consequences which actually occur in some branch) and wi is the weight of the
consequences with value Vi , i. e. the sum of the squared moduli of all branches
in which payoffs of value Vi are made.
As with Deutsch’s own proof, we suppose P(x) = x by default, and hold fixed
b to be measured: this allows us to write h|ψii for h|ψi , X,
b Pi.
the observable X
In this case, we also write EU (|ψi) for EU (G).
42
Stage 1 Any game is characterised by the distinct numerical values of the payoffs given in its branches, and their weights.
The GET tells us that any game is characterised by the distinct consequences
and their weights. From Dominance, it follows that two games which differ only
by substituting some of the payoffs for equal-valued payoffs are equivalent, and
two payoffs are equivalent iff they have the same numerical value.
b V(|ψi) =
Stage 2 If G is an equally-weighted superposition of eigenstates of X,
EU (ψ).
Without loss of generality, suppose |ψi = (1/N )(|λ1 i + · · · + |λN i).
Let π be an arbitrary permutation of 1, . . . , N , and define Pπ by Pπ (xi ) =
xπ(i) . Then by act additivity,
X
X
X
b Pπ ) = V(|ψi , X,
b
xi
(40)
V(|ψi , X,
Pπ ) = (n − 1)!
π
π
i
P
since π Pπ is just the constant payoff function that gives a payoff of (n −
1)!(x1 + · · · + xN ) irrespective of the result of the measurement.
b Pπ i is a game in which each consequence
But each of the n! games h|ψi , X,
xi occurs with weight 1/N . Hence, by the GET, all have equal value, and that
value is just V(|ψi). Thus, n!V(|ψi) = (n − 1)!(x1 + · · · + xN ), and the result
follows.
P
Stage 3 If |ψi =
i ai |λi i, where the ai are all rational, then V(|ψi) =
EU (|ψi).
Any such state may be written
√ X√
|ψi = (1/ N )
mi |λi i ,
(41)
i
P
where the mi are integers satisfying
i mi = n. Such a game associates a
weight mi /N to payoff xi .
But now consider an equally-weighted superposition |ψ ′ i of n eigenstates of
b where a payoff of x1 is given for any of the first m1 eigenstates, x2 for the
X
next m2 , and so forth. Such a game is known (from stage 2) to have value
(1/N )(m1 x1 + · · · + mN xn ) ≡ EU (|ψi). But such a game also associates a
weight mi /N to payoffs of value xi , so by stage 1 we have h|ψii ≡ h|ψ ′ ii and
the result follows.
Stage 4 For all states |ψi which are superpositions of finitely many eigenstates
b V(|ψi) = EU (|ψi).
of X,
By the GET, it is sufficient to consider only states
X
αi |λi i
|ψi =
i
43
(42)
b
with positive real α1 . Let |µi i, (1 ≤ i ≤ N ), be a further set of eigenstates of X,
orthogonal to each other and to the |λi i and with eigenstates yi distinct from
each other and all strictly less than all of the xi (that we can always find such
a set of states, or reformulate the problem so that we can, is a consequence of
GET, or SET if preferred). For each i, 1 ≤ i ≤ N , let ani be an increasing series
of rational numbers converging on (αi )2 , and define
Xp
Xp
|ψn i =
(43)
ani |λi i +
ai − ani |µi i .
i
i
It follows from stage 3 that V(|ψn i) = EU (|ψn i), and from Dominance
that for all n, V(|ψi) ≥ V(|ψn i). Trivially limn→∞ EU (|ψn i) = EU (|ψi), so
V(|ψi) ≥ EU (|ψi). Repeating the construction with all the yi strictly greater
than all the xi gives V(|ψi) ≤ EU (|ψi), and the result follows.
5.4
Hidden variables
As mentioned in section 1, Deutsch’s argument (and, as will be seen, my generalisations of it) rely essentially on the assumption that the Everett interpretation
is correct. It may then be instructive to see how the argument fails in one
particular set of non-Everett interpretations: those involving ‘hidden variables’,
such as the de Broglie-Bohm theory (Bohm 1952; Holland 1993).
Recall that in a hidden-variable theory, the physical state of a system is
represented not just by a Hilbert-space vector |ψi, but also by some set ω of
hidden variables, so that the overall state is an ordered pair h|ψi , ωi. (In the
de Broglie-Bohm theory, for instance, ω is the position of the corpuscles.)
The Deutsch argument and its generalisations rely on the invariance of the
state under certain unitary transformations, and to apply the argument to a
hidden-variable theory we need to know how these transformations act on the
hidden variables. This will in general depend upon the hidden-variable theory
in question; however, if the hidden variables are intended to represent spatial
positions of particles (as is generally the case, and in particular is true for de
Broglie-Bohm corpuscles) we can specialise to position measurements and to
spatial translations and reflections, whose effects upon corpuscle positions are
clear.
Suppose, in particular, that we consider a measurement of the spatial position of √
a particle in one dimension, and assume that the quantum state is
|ψi = (1/ 2)(|xi + |−xi), where |xi and |−xi are eigenvectors of position with
eigenvalues x and −x respectively. Deutsch’s argument relies on the fact that
this state is invariant under reflections around the origin. If the argument is
to generalise to hidden variables, we will then require that their positions are
also invariant under reflection — which forces them to be located at the origin.
However, since the hidden variable is supposed to represent the actual position of the particle, it must be located either at +x or −x, since these are the
only possible results of a position measurement on state |ψi; it follows that the
Deutsch argument cannot apply to such systems.
44
We could, of course, try to get round this problem by considering a probability distribution over hidden variables and requiring the distribution to be
symmetric. Fairly clearly, this forces a distribution assigning probability 0.5 to
both +x and −x. The Deutsch argument can now be applied, and yields the
unedifying conclusion that if the particle is at position +x 50 % of the time, it
is rational to bet at even odds that it will be found there when measured.
6
Going beyond additivity
We have seen that, in the classical case, additivity is ultimately not required as
an assumption, at least in deriving probabilities. In this section we will show
that additivity may be dispensed with in the quantum case also, and replaced
by a purely qualitative set of axioms.
6.1
Quantum versions of the Savage axioms
The postulates which we shall need to prove the improved version of Deutsch’s
result are modelled closely on Savage’s axioms (described in section 2.6). They
are as follows:
QS0: Act availability The set of acts is AQ .
QS1: Value Preference: Acts There is a weak asymmetric order ≻ on the
set A, which satisfies measurement neutrality. (As usual ≻ induces a weak
order on the set of consequences, via the constant acts.)
QS2 : Non-triviality There are at least two consequences c, d with c ≻ d.
b P1 i and G2 = h|ψi , X,
b P2 i,
QS3: Sure-Thing Principle In two games G1 = h|ψi , X,
if the payoff functions P1 and P2 agree on some subset R of the spectrum
b then the preference order between G1 and G2 does not depend on
of X,
what actual value is taken by P1 (and thus P2 ) on R.
QS4: Dominance Let G1 and G2 be as above. If P1 (s) P2 (s) for all measurement outcomes s then G1 G2 ; if in addition P1 (s) ≻ P2 (s) for some
s which actually occurs in some post-measurement branch, then G1 ≻ G2 .
QS5: Structural assumption See below. This axiom will play the role of
Savage’s structural axiom S5. It turns out that there are various choices
for QS5; these will be discussed later.
Of these postulates:
• QS0 and QS1 are essentially the same as Deutsch’s D0 and D1.
• QS2–QS4 are almost verbatim copies of Savage’s S2–S4.
• QS5 will play the role of Savage’s structural axiom S5. It turns out that
there are various choices for QS5; these will be discussed later.
45
• The only Savage axioms which is not represented here are the probability
axiom S5, which ensures that all events have comparable probability, and
the boundedness axiom S7, which allows the theory to be extended to
games with infinitely many distinct consequences. We will see that S5 can
be derived in the quantum case; whether or not a quantum version S7 is
needed depends on the form of QS5, as discussed below.
6.2
Probabilities for quantum events
Suppose we consider a given measurement M ∈ MQ , specified by the state
b Let SM be the set of
|ψi to be measured and the measured observable X.
b spectrum which are possible outcomes of the measurement:
all elements of X’s
that is, all eigenvalues with at least one associated eigenvector |λi such that
hψ|λi =
6 0. Then the events for M will be all subsets of SM , and C̃ will denote
C’s complement in SM . The weight of an event will be defined in the obvious
way, as the sum of the weights of all branches contributing to that event.
b Pi, such that P equals x on C
Define a bet on C as a game G = h|ψi , X,
and y on C̃, where x and y are consequences and x ≻ y. Such a bet will be
denoted hM, C, x, yi.
We will define a qualitative probability measure on events as follows: given
any two events C and C ′ ,
• C is more probable than C ′ (written C ≻ C ′ ) if and only if hM, C, x, yi ≻
hM ′ , C ′ , x, yi for all x, y such that x ≻ y.
• C is equiprobable to C ′ (written C ≃ C ′ ) if and only if hM, C, x, yi ≃
hM ′ , C ′ , x, yi for all x, y such that x ≻ y.
Note that this definition allows comparison of events associated with different
measurements; note also that “C ≃ C ′ ” is not synonymous with “neither C ≻ C ′
nor C ≺ C ′ ”. In fact, we have not yet proved that an arbitrary pair of events
are comparable in respect of probability at all (recall that this is an axiom —
S5 — in the Savage framework).
The probability relations inherit obvious properties from the order on acts:
specifically, ≻ is a partial ordering on events, and ≃ is an equivalence relation.
We will now prove:
Probability theorem:
1. All events are comparable in probability, with C ≻ C ′ iff C has
strictly greater weight than C ′ .
2. There exists one and only one function Pr on the set of all
events such that
(a) Pr(C) > Pr(C ′ ) iff C ≻ C ′ ;
(b) When restricted to the events of a given measurement M ,
Pr is an additive measure: that is, for disjoint events C, D ∈
SM , Pr(C ∪ D) = Pr(C) + Pr(D);
46
(c) For any M , Pr(SM ) = 1.
That function is the weight function: Pr(C) equals the weight
of C.
To begin, note that any bet hM, C, x, yi, where C has weight w, is a game
where consequence x occurs with weight w and y occurs with weight (1−w). By
the GET, this is sufficient to specify the game completely for decision-theoretic
purposes, so we can write hw, x, yi for any such bet without relevant ambiguity.
It follows immediately that two events are equiprobable whenever they have
equal weight.
√
√
′
′
√ Now suppose 0 < w < w ≤ 1, and set |ψi = w |λ1 i + w − w |λ2 i +
1 − w′ |λ3 i, where x1 , x2 , x3 are all distinct. Define P by
and P ′ by
P(x1 ) = x, P(x2 ) = y, P(x3 ) = y,
(44)
P ′ (x1 ) = x, P ′ (x2 ) = x, P ′ (x3 ) = y.
(45)
b P ′ i ≻ h|ψi , X,
b Pi. But h|ψi , X,
b Pi
Then by Dominance, if x ≻ y then h|ψi , X,
′
′
b
is a realisation of the bet hw, x, yi, and h|ψi , X, P i realises hw , x, yi. Hence any
event of weight w′ is more probable than one of weight w; this completes the
proof of part 1.
The proof of part 2 proceeds in a similar way to my alternative proof of
Deutsch’s result: we begin with equally weighted superpositions and proceed
successively to rationally weighted superpositions and general superpositions.
Define a probability function as any function on the set of events satisfying
(a)-(c) of part 2 of the probability theorem; let Pr be an arbitrary probability
function, and let W be the weight function, assigning to each event its weight.
In view of part 1, we must have Pr(C) = Pr(C ′ ) iff W (C) = W (C ′ ).
Lemma 1 Pr(C) = W (C) is a probability function.
Given part 1, it is easy to verify that Pr(C) = W (C) satisfies (a)-(c) of part
(2).
Lemma 2 If SM = x1 , . . . xN , all distinct, and if each state in SM has the
same weight, then Pr({xi }) = W (xi ) = 1/N.
Since all the states have the same weight, each event {xi } must have the
same probability. Since Pr is required to be additive and Pr(SM ) = 1, this
forces P r({xi }) = 1/N .
Lemma 3 Any event C with rational weight satisfies Pr(C) = W (C).
Let W (C) = K/N . Consider any measurement M for which SM = {x1 , . . . , xN }
is a set of equally-weighted states; then by additivity of Pr, Pr({x1 , . . . , xK }) =
K/N. But W ({x1 , . . . , xK }) = K/N = W (C), and Pr is the same for any two
events with the same weight.
47
Lemma 4 Pr(C) = W (C) is the unique probability function.
In view of lemmas 1–3, all that is left to prove is that Pr(C) = W (C) on
irrationally-weighted events. Let C be an arbitrary event, with weight w; let
{wn } be an increasing sequence of rational numbers convergent on w, and {Cn }
a sequence of events with W (Cn ) = wn . Then clearly we must have Pr(C) ≥
Pr(Cn ) for all n, and hence Pr(C) ≥ W (C). Repeating with a decreasing
sequence shows that Pr(C) ≤ W (C), and the result follows.
It is interesting to compare the Probability Theorem with the analogous result in Savage’s framework. There too, it is provable that an agent’s preferences
determine a unique probability measure on the space of events, and in fact the
method of proof is pretty similar: first “more probable than” is shown to order
the events, then the space of events is carved into arbitrarily many equiprobable
(or almost equiprobable) sub-events and these are used to show that there is a
unique probability function on th events compatible with this ordering.
There are important differences, however:
1. In the quantum approach, rather fewer axioms of pure rationality are
needed. We can dispense with the assumption that all events are comparable in respect of probability (S5), and with the sure-thing principle
(S3).
2. There is also no need to use Savage’s structural assumption (S6), which
he requires to show that the set of events can be arbitrarily divided up.
In fact, for any given quantum measurement the set of states is finite and
so the events certainly cannot be carved arbitrarily finely; however, measurement neutrality lets us replace any measurement with an equivalent
one which has more events, which is an adequate substitute. The rich
structure of QM excuses us from a need to postulate this structure at the
decision-theoretic level.
3. Most importantly, all rational agents must agree on their probabilistic assignments in the quantum case, whereas there is scope in Savage’s system
for many different assignments.
6.3
Choices of structural axiom
The next step in Savage’s proof of the EUT (having obtained a unique probability measure) is to show that the von Neumann-Morgenstern axioms VNM0–3
are satisfied (after which the EUT follows directly from von Neumann’s and
Morgenstern’s result.) To do so Savage again needs to use his structural axiom
S6 (which we avoided using in proving the Probability Theorem), and now we
too will be forced to use some analogous axiom.
However, the fact that we did not need S6 for discussions of probability
suggests that it may be too strong for our purposes, and that it might be
possible to weaken it. Recall (section 2.6) that S6 does four things for Savage:
it guarantees that the space of events is continuously divisible, it rules out
48
infinitesimal probabilities, and it requires that no two events differ infinitely or
infinitesimally in value. It is only the latter two properties which we need, since
we have probability in hand; this suggests the following variant of S6.
QS5a: Comparability of acts Given any three games G1 , G2 , G3 with G1 ≻
G2 , there is some G1 − G3 bet which is preferred to G2 , and some G2 − G3
bet to which G1 is preferred.
(Here a G − G ′ bet has the obvious meaning: a bet on some event A such
that we get to play G if A obtains, and otherwise play G ′ .) In the presence of
measurement neutrality, Q5a obviously entails S6.
There is, however, a completely different strategy available, which again
makes use of the rich structure of quantum mechanics to reduce structural constraints on decision-making. This strategy replaces QS5a with
QS5b: Stability Suppose G1 and G2 are games with G1 ≻ G2 ; then this preference is stable under arbitrarily small perturbations of the states being measured in the two games. Symbolically, this is to say that, if
b 1 , P1 i ≻ h|φi , X
b 2 , P2 i then there exists some ǫ > 0 such that, if
h|ψi , X
|ψ ′ i and |φ′ i are any states satisfying | hψ|ψ ′ i | > 1−ǫ and | hφ|φ′ i | > 1−ǫ,
b 1 , P1 i ≻ h|φ′ i , X
b 2 , P2 i.
then h|ψ ′ i X
Q6b turns out to be just as effective as Q6a in ruling out infinitesimal or
infinite values. It is clearly not an axiom of ‘pure’ decision theory: it makes
essential reference to the quantum mechanics of the decision problem under
consideration. This being the case, why use it instead of the ‘pure’ structure
axiom QS5a?
Partly, QS5b is preferable because it allows a simple extension of the expectedutility rule to infinite games, whereas QS5a has to be supplemented to do so
(see section 6.5).
Most importantly, though, QS5b admits of a very reasonable justification
— more reasonable, perhaps, even than QS5a. For without it, it would not
be possible for any agent without Godlike powers to act in accordance with
his preferences. After all, without it then the agent would have to prepare the
state to be measured — not to mention the measuring device — with infinite
precision. Any finite precision, no matter how good, will fail to tell the agent
which act to prefer unless QS5b holds. For instance, suppose QS5b fails to
hold for two acts f , g (where f ≻ g) and suppose f involves measuring the z−
component of spin of some spin state |ψi. Preparing that state will presumably
involve aligning an initial spin with some magnetic field (or something similar)
and any finite error in the alignment of that field will lead to finite errors in the
preparation of |ψi.
But given the failure of QS5b, for any ǫ > 0 — however small — there will
exist some |ψ ′ i with | hψ|ψ ′ i | > 1 − ǫ — but where measuring |ψ ′ i instead of |ψi
is not preferred to g. This means that even if in principle the agent knows that
he would prefer f to g, he can never know whether any given act resembling f
is preferred to g.
49
This shows that at the very least an agent with preferences violating QS5b
would be unable to use decision theory as a practical guide to action. Either
such an agent is reduced to catatonia, or he must adopt some secondary theory of decision-making which is practically implementable, even if (as it must)
it conflicts in places with his bizarre “real” preferences; this secondary theory
will have to obey QS5b, and arguably better describes the agent’s ‘real’ preferences between acts than his purely verbally expressed preference for (say)
measurements whose outcomes have rationally-valued weights over those with
irrationally-valued weights. In fact, if we follow Lewis’s advice and treat preferences as purely determined by dispositions to action, then violation of QS5b
by an agent is not just pathological, but physically impossible.
6.4
Expected utilities for quantum events
Savage’s proof of VNM0–3 is straightforward but rather lengthy. I will only
sketch it here, as well as the minor variations to it required in the quantum
case; see Savage (1972) for the full details.
Savage’s Step 1 Show that two acts are value-equivalent whenever they assign
the same probability as each other to each consequence. (In other words, f and
f ′ are value-equivalent whenever there exist partitions Ci , Ci′ of the state space
such that P (Ci ) = P (Ci′ ) and f (x) = f ′ (x′ ) whenever x ∈ Ci and x′ ∈ Ci′ .)
This is Savage’s Theorem 5.2.1. In quantum mechanics, the equivalent result is
that two acts are value-equivalent whenever they each assign weights p1 , . . . , pn
to each of n consequences c1 , . . . , cn ; this follows immediately from the GET
and the Probability Theorem.
Savage’s Step 2 Define a gamble as an equivalence class of acts each of which
assign the same probability as one another to each consequence; denoteP
a gamble f which assigns
probabilities
p
,
.
.
.
,
p
to
consequences
c
,
.
.
.
,
c
by
1
n
1
n
i p i ci .
P
P
P
Define a mixture j σj fj of gambles fj = i pij cij as the gamble ij, (σi pij )cij ,
and show that such mixtures always exist (this is a consequence of the structural
axiom S6).
These definitions go through unchanged
in the quantum case; to show that mixP √
tures always exist, let |ψi = ij σi pij |λij i where the |λij i are eigenstates of
b with eigenvalues xij , all discrete; let P be a payoff scheme assigning conseX
b Pi instantiates the mixture
quence cij to measurement result xij . Then h|ψi , X,
of gambles.
Savage’s Step 3 Show that, if f, g and h are gambles and 0 < ρ ≤ 1, then
ρf + (1 − ρ)h ≻ g + (1 − ρ)h iff f ≻ g.
This is Savage’s Theorem 5.2.2, and its proof makes essential use of S6. The
quantum proof is identical save for the substitution of either QS5a or QS5b for
S6; as it is moderately cumbersome it will not be given here (see Savage 1972,
p. 72).
50
Savage’s Step 4 Show that, if f1 ≺ f2 and f1 g f2 , then there is one and
only one ρ such that ρf1 + (1 − ρ)f2 ≃ g.
This is Savage’s Theorem 5.2.3; it is an easy consequence of step 3 and S6.
Though the quantum proof is virtually identical (again substituting QS5a or
QS5b for S6) I will give it to illustrate the sort of use that is made in the theory
of QS5a/QS5b.
From the quantum version of step 3 and the principle of Dedekind cuts, we
know that there is one and only one ρ such that
σf1 + (1 − σ)f2 ≺ g if σ > ρ;
(46)
σf1 + (1 − σ)f2 ≻ g if σ < ρ.
(47)
It follows that no number except possibly ρ can satisfy the equivalence which
the theorem requires. Suppose for contradiction that ρ does not in fact satisfy
it: without loss of generality assume
ρf1 + (1 − ρ)f2 ≻ g.
(48)
We can show this is contradictory via QS5a, which entails the existence of some
λ > 0 such that
λf1 + (1 − λ)(ρf1 + [1 − ρ]f2 ) ≻ g,
(49)
but this is equivalent to
(ρ + λ[1 − ρ])f1 + (1 − λ)(1 − ρ)f2 ≻ g,
(50)
in contradiction with (47).
Alternatively, note that an arbitrarily small perturbation of the state being
measured will change ρ to some ρ′ with ρ′ > ρ. By QS5b, there will be some
sufficiently small perturbation of this sort which preserves the preference (48),
but this is in contradiction with (47).
Savage’s Step 5 Steps 3 and 4 establish, respectively, the von Neumann-Morgenstern
axioms VNM2 and VNM3; VNM0 and VNM1 are immediate consequences of
S0 and S1. As such, the von Neumann-Morgenstern theory of utility can be
applied, showing that to each consequence can be associated a unique numerical value such that for all games Gi with finitely many distinct consequences,
G1 ≻ G2 iff EU (G1 ) > EU (G2 ).
This result goes through unchanged in the quantum case: we have therefore
established the expected-utility rule in that case also.
6.5
Infinite games
We have now proved the validity of the expected-utility rule (and with it the
identification of probabilities with weights) for all games with finitely many
51
distinct consequences — and, in particular, for all games played in a finitedimensional Hilbert space. This is as far as Deutsch went in his proof, and
possibly as far as we need go: the Bekenstein bound offers powerful reasons
why the Hilbert space of the Universe must be finite, and more prosaically the
state space of a finite-volume system (such as a measurement device, or indeed
a human brain) whose energy is bounded above must also be finite.
Nonetheless, we can fairly straightforwardly extend the expected-utility rule
from finite to infinite games. In Savage’s decision theory, this is done in two
steps: firstly S7 is used to show that the value function V is bounded, and then
this is in turn used (again via S7) to extend the rule. We can take this approach
directly over to the quantum case by adding S7 to the quantum axioms (as
QS6).
However, it is interesting to note that if we adopt the stability axiom QS5b
in place of QS5a, we can prove the extension to infinite games without any
need to add S7 to the axioms; in the rest of this section we will prove this.
b
Infinite games are only possible if it is possible to measure some operator X
with infinitely many distinct eigenstates; we will assume henceforth that some
such measurement is contained within MQ .
Lemma 1 The set V(C) is bounded.
Proof: we will concentrate on proving that V(C) is bounded above; proving it
to be bounded below proceeds in an essentially identical way.
Suppose for contradiction that V(C) is unbounded above, and let c1 , c2 , . . . be
a sequence of consequences such that ci ≻ cj for i > j and limi→∞ V(ci ) = +∞.
b with distinct eigenvalues
Let |λ1 i , |λ2 i . . . be a sequence of eigenstates of X
x1 , x2 , . . . and set P(xi ) = ci .
b Pi ≻ h|λ1 i , X,
b Pi, it follows from QS5b that there exists
Since h|λ2 i , X,
b
b
some ǫ > 0 such that h|λ2 i√, X, Pi ≻ h|ψi
√, X, Pi whenever | hλ1 |ψi | > 1 − ǫ.
In particular, if |ψi = 1 − ǫ |λ1 i + ǫ |λi i, for any i, then this condition is
satisfied. By the expected-utility principle for finite-outcome games, it follows
that
(1 − ǫ)V(c1 ) + ǫV(ci ) < V(c2 )
(51)
for fixed ǫ and all i, which contradicts the unboundedness of the set of consequences.
Theorem 6.1 The expected-utility rule holds for all games, finite and infinite.
Proof: Using measurement neutrality, any game G can be mapped onto some
b Pi, where
game of the form h|ψi , X,
b is some fixed non-degenerate observable, and P a fixed payoff scheme
• X
for that observable.
b consist of the doubly infinite sequence
• The eigenstates of X
. . . , |λ−2 i , |λ−1 i , |λ0 i , |λ1 i , |λ2 i , . . .
52
(52)
with |λi i having eigenvalue xi , and two additional eigenstates |λ+ i and
|λ− i, with eigenvalues x+ and x− respectively.
• P satisfies P(xi ) P(xj ) whenever i ≥ j.
• V[P(x+ )] equals the upper bound of V, and V[P(x− )] equals the lower
bound.
P
• |ψi = +∞
i=−∞ αi |λi i. (Note that the two ‘extra’ eigenstates |λ± i are not
included in this sum.)
b Pi.
As such, any game can be referred to just by its state: h|ψii ≡ h|ψi , X,
For any game G, define EU (G) as the expected utility of G (the boundedness
of V guarantees that this is well-defined even for infinite games). In particular,
EU (|ψi) denotes the expected utility of h|ψii.
Now for each n, define states |ψn+ i and |ψn− i by


sX
X
±
ψ =
αi |λi i + 
(53)
|αi |2  |λ± i ;
n
|i|<n
|i|≥n
note that limn→∞ EU (|ψn± i) = EU (|ψi).
By QS3 (Dominance) we have, for all n,
hψn+ i h|ψii hψn− i.
(54)
But the games h|ψn± ii have finitely many distinct consequences, and so the
expected-utility rule applies to them. It follows that h|ψii ≻ G whenever G is
some finite game with EU (|ψn− i) > EU (G) for any n, and hence that h|ψii ≻ G
whenever EU (G) < EU (|ψi); similarly, G ≻ h|ψii for any finite game G with
EU (G) > EU (|ψi).
Now, by choosing appropriate weights we can always construct some finite
game G with EU (|ψi) = EU (G); if we can prove that h|ψii ≃ hGi then the proof
will be complete.
Assume for contradiction that G ≃
/ h|ψii — without loss of generality we may
assume h|ψii ≻ G. Then, by the stability postulate QS5b, there exists some ǫ > 0
such that h|ψ ′ ii ≻ G whenever | hψ|ψ ′ i | > 1 − ǫ. There will always be some |ψn− i
satisfying this condition, so we have h|ψn− ii ≻ G. But EU (|ψn− i) < EU (G), in
contradiction to the expected-utility rule. Since this rule is known to hold for
all finite games, the contradiction is established.
7
Gleason’s Theorem
Before going on to discuss the implications of the proofs of sections 5–6, we
should consider a possible alternative to the whole approach. It has been argued
(notably by Barnum et al (2000) in their recent criticism of Deutsch) that
Gleason’s Theorem in any case gives us all we could want in the way of a
53
derivation of the quantum probability rule. In this section I will attempt to
sketch a proof of the probability rule using Gleason’s theorem, and then argue
that it is in a number of ways less satisfactory than the approaches of sections
5–6.
7.1
Deducing probabilities from Gleason’s Theorem
Gleason’s theorem itself, a piece of pure mathematics, states the following:
Let H be a Hilbert space of dimension > 2. Let f be any map from
b i } is any set
the projectors on H to [0, 1] with the property that, if {P
P
P b
b
b
of projectors satisfying i P i = 1, then i f (P i ) = 1. Then there
b ) = Tr(P
b ρ)
exists a unique density operator ρ on H such that f (P
b
for all projectors P .[Gleason 1957]
The theorem is remarkable in its generality: notice in particular that no
restriction of continuity has been placed on f (although of course the conclusion
of the theorem entails that f must in fact be continuous).
If we are trying to construct some sort of instrumentalist interpretation of
quantum mechanics, Gleason’s Theorem is little short of a panacea. Such an
interpretation needs only to presume
1. that physically possible measurements on a system are represented by the
algebra of self-adjoint operators on some Hilbert space (and, in particular,
measurements giving YES-NO answers are represented by projections);
2. that the theory should assign probabilities to each possible measurement
outcome; and
b and B
b are commuting self-adjoint operators, then the proba3. that if A
b does not depend upon
bility of getting a given result upon measuring A
b
whether B is measured simultaneously. (This assumption is called noncontextuality, and guarantees that probabilities need be assigned only to
individual projectors, rather than jointly to sets of them).
Then Gleason’s Theorem implies that the only possible probability choices are
those represented by a (pure or mixed) quantum state. This even allows the
instrumentalist to deny all reality to the quantum state: quantum states can
simply be ways of codifying an agent’s beliefs about the outcomes of possible
measurements.
7.2
Applying Gleason’s Theorem to the Everett interpretation
However, we are not engaged in the project of constructing an instrumentalist
interpretation of QM. The Everett interpretation is robustly realist about the
quantum state right from the beginning, and so applying Gleason’s Theorem to
it is not in any way needed to justify the existence of the state.
54
Nonetheless, might it be possible to use the Theorem to justify the quantum probabilities? Only if it can be established somehow that the concept of
probability is a necessary consequence of decision-making in situations involving
quantum uncertainty, for Gleason’s theorem cannot be applied at all until we
can justify the need to assign probabilities to branches.
We have already seen that this justification can be given if we assume QS0–2
and QS4 — but we have also seen that this leads almost immediately not just
to the existence of probabilities, but to the quantum probability rule itself. It
follows that use of Gleason’s Theorem would only be of use if it can somehow
enable us to weaken QS0–2 or QS4.
The only plausible way in which this could work, as far as I can see, would
be via a modification of QS0. That axiom requires the agent to have preferences
defined over a rather large class of games and Gleason’s theorem suggests that
that class could be modified somewhat.
A possible axiom scheme that could implement such an idea might be:
G0: Act availability The set of available acts includes all possible measurements of self-adjoint operators on the Hilbert space H of some fixed quantum system with Dim(H) > 2, as well as some set of bets on the outcomes
of these measurements.
G1: Transitive preferences The agent has some preference order ≻ on pairs
of acts, which is a weak ordering of the set of acts (defining, as usual,
a weak ordering of the set of consequences) and satisfies measurement
neutrality. (i. e. , as for QS1).
G2: Non-triviality (As for QS2).
G3: Sure-Thing Principle (As for QS3).
G4: Dominance (As for (QS4).
G5: Probability ordering Any two events are comparable in probability (in
the sense of section 6.2).
G6: Structure axiom (As for QS5, i. e. either QS5a or QS5b).
These axioms are quite similar to QS1–5; note, however, that
• The set of games considered is rather different: now we need consider only
one physical system, but we need to be able to realise all measurements
on it (or, which is equivalent, perform all unitary transformations). This
means that measurement neutrality leads to rather different results (in
particular, the proof of the Grand Equivalence Theorem now fails).
• As a consequence of losing the Grand Equivalence Theorem, it is necessary
to postulate that all events are comparable in respect of probability (a la
S5) rather than deriving it.
55
The proof of the probability theorem would now go as follows: Firstly, we
know that all events are comparable in respect of probability. This tells us that
probability defines a weak ordering on the set of all projectors on H. With
the aid of Dominance, we can also prove that the projector onto the physical
state |ψi of the system is not less probable than any other projector, as follows:
b with eigenvalue xa ; let P be a
let |ψi be an eigenstate of some operator X
b
payoff scheme for X allocating payoff x to xa and y to all other eigenvalues,
b Pi realises an x, y bet on xa .) Then since payoffs
with x ≻ y. (Hence h|ψi , X,
are not counterfactually defined, and payoff x is guaranteed to occur, we can
replace P by a payoff scheme which always gives x. By Dominance, the resultant
game is at least as probable as any other x, y bet, and the result follows by the
definition of the ‘more probable than’ relation.
Suppose it were the case that the qualitative probability ordering could be
represented : that is, that there is some function f from the projectors to [0, 1]
which restricts to a probability measure on the spectrum of each observable
b 1 ) > f (P
b 2 ) iff P
b1 ≻ P
b 2 . Then Gleason’s
operator and which satisfies f (P
b) =
theorem would imply the existence of some density operator ρ such that f (P
b ρ).
Tr(P
(It may be helpful, here, to observe that measurement neutrality implies
that any representation of the preferences is noncontextual. This can be seen
b 1 be a projector, and
most directly from the Grand Equivalence Theorem: let P
suppose we consider a bet in which we measure a state with respect to some
b 1 , getting a reward if and only if the
commuting set of projectors including P
b 1 occurs. By the Grand Equivalence Theorem, such
outcome corresponding to P
a bet is specified completely (for decision-theoretic purposes) by the weights
b 1 |ψi and 1 − hψ| P
b 1 |ψi — the details of what is measured concurrently
hψ| P
b
with P 1 are decision-theoretically irrelevant.)
We would not yet have shown that the quantum state generating f was the
physical quantum state, |ψi; this, however, would be easy. For the considerations above tell us that the probability of obtaining xa upon measuring the obb must be no lower than the probability of any other event — and hence
servable X
must equal 1. The only quantum state ρ satisfying Tr(ρ |ψi hψ|) ≡ f (|ψi hψ|) = 1
is of course |ψi hψ| itself.
This having been established, the remainder of a Gleason’s Theorem-motivated
derivation of the EUT would be able to proceed much as in section 6.4.
All this relies, however, on our ability to show that any qualitative probability possesses a representation, and I am not aware of any proof of that fact.
Effectively, Gleason’s theorem provides a uniqueness proof for probability but
not an existence proof, and for all I know it is necessary to introduce additional
axioms of a structural character to ensure this.
One way of so doing would be to use the additivity strategy to define the
value function on consequences in advance of considering probability: then, recall, bets can be used to define probabilities. A set of axioms implementing this
strategy would be virtually identical to D0′ –D4′ , differing only in the replacement of the act-availability axiom D0′ with G0.
56
7.3
Disadvantages of the Gleason approach
How does the derivation of the probability rule from G0 and G2–G4 compare
with those given in sections 5–6? The most obvious disadvantage of the derivation via Gleason’s Theorem is of course that it is incomplete, with the existence
of any numerical representations of the probability ordering being unproven.
(Granted, adding the assumption of Additivity solves this problem, but we have
already seen that that assumption is implausibly strong).
Let us assume, though, that the existence of a representation can in fact be
proved from the G axioms without adding any over-strong structure axioms. In
that case, the assumptions made in this section’s derivation of the probability
rule seem pretty much on a par with the Savage-inspired derivation of section
6. Granted, in this section we needed to assume explicitly (G5) that events are
comparable in respect of probability, whereas this could be deduced explicitly
in section 6; but G5 is a highly reasonable-sounding axiom of pure rationality,
transferred mutatis mutandis from the classical Savage axioms. The other main
difference is the set of acts considered: section 6 requires systems of all dimension
to be considered, whereas the present section required only a fixed dimension
but also required that all unitary transformations be considered performable.
Neither set of acts seems obviously more reasonable than the other, though:
both are clearly idealisations, but fairly reasonable ones (especially since we
only require agents to have well-defined preferences between such acts and not
necessarily to be able to perform them; I have a definite preference between
exile on Saturn and being appointed World President, though it is certainly not
in my power to bring either consequence about!)
If there is reason to prefer a proof modelled on classical decision theory
to one based around Gleason’s theorem, it probably lies more in the proofs
themselves than in the axioms required. Here it seems that the former proof
has a major advantage in brevity and simplicity: the proof of the probability
theorem presented in section 6.2 takes barely a page, whereas even the proof of
Gleason’s Theorem is long and complicated, and we have seen that this would
be only one component of a proof based on the axioms of this section.
Furthermore, conceptually speaking it is important to remember that Gleason’s Theorem is here being used not as an alternative to a decision-theoretic
approach, but as a component of it. It is not the case that simple invocation
of Gleason’s Theorem, unburdened with considerations of a decision-theoretic
character, is sufficient to resolve the Everettian probability problem.
8
Quantum probabilities: a discussion
In this section I will consider the implications of the proofs just given. I will
show that they guarantee that quantum weights satisfy the requirements (discussed in section 2.7) placed by Papineau on an adequate analysis of chance,
and strongly support (though they do not quite entail) a quantum-mechanical
version of Lewis’s Principal Principle. I will also give a brief discussion of some
57
implications of this new Principle.
8.1
The decision-theoretical link
One of Papineau’s two links between objective chances and non-probabilistic
facts was that it is rational to use objective chances as a guide to our actions. It
might seem that we have justified this already: if I am offered a bet where the
payoff depends on the outcome of a measurement on a known state, the whole
point of our discussion of decision theory was to prove that the correct strategy
was to calculate the expected utility of the bet using the quantum weights as
probabilities.
It is not quite so straightforward as that, though. For suppose that the
measurement has already happened — in a sealed box on the other side of
the lab, say. Even though the box is sealed, decoherence will still have led to
branching, and there will be (at least) one branch for each possible outcome of
the measurement. My uncertainty is now simple ignorance: determinately I am
in one branch rather than another (for all that there are copies of me in the
other branches) but I do not know which.
It is surely part of the quantum probability rule that the right strategy to
adopt here is exactly the same as if the measurement had not yet occurred:
namely, maximise expected utility with respect to the weights of the various
measurement outcomes. This can, in fact, be deduced from our results so far,
provided we accept a further principle of rationality, called the reflection principle by van Fraassen (1984).
The reflection principle: If, at time t, I decide rationally to pursue a certain
strategy at a later time t′ , and if I gain no new information relevant to
that strategy between times t and t′ , then it is rational not to change my
choice of strategy at t′ .
Like all axioms of pure rationality, this seems very obvious (and admittedly,
like virtually all of them. it has been challenged; see, e. g. , (Elga 2000)). We
can use it to deduce the post-measurement strategy very easily, as follows: if
you had decided which strategy to adopt before the measurement, you would
have opted for maximising expected utility with respect to quantum weights.
The reflection principle says that you should adopt the same strategy even after
the measurement, as ex hypothesi you have gained no information at all about
the results of the measurement.
With this additional result in place, it seems that we have satisfied Papineau’s requirement that a theory of probability explain probabilistic decisionmaking.
8.2
The inferential link
Papineau’s other requirement was that probabilities could be estimated via
frequencies, and we will show the validity of this in the context of quantum
games. For simplicity, consider only a two-state system, prepared in some state
58
|ψi = α |0i + β |1i and measured in the |0i , |1i basis. We have a choice of two
bets: either get a reward of value x if the measurement yields “0”, and of value
y if it gives “1”; or get a reward of value 0 with certainty.
Obviously (given our derivation of the EUT in the quantum case) the correct
strategy is to take the first bet if and only if |α|2 x + |β|2 y > 0; but what if we
don’t have a clue about the values of α and β? Then it seems that (absent
further information) we cannot choose which strategy to follow.
However, the situation changes substantially if we have access to a large
number of copies of |ψi and can measure each copy in the |0i , |1i basis. The
classical strategy would then be something like: measure all N copies; if we get
M results of “0”, then take the bet provided that (M/N )x + ([N − M ]/N )y > ǫ,
for some small ǫ. (This is equivalent to using the relative frequencies as estimates
of the probabilities, with ǫ set non-equal to zero to hedge against the fluctuations
possible in the estimate.)
How do we fare if we adopt this strategy in quantum mechanics? The combined weight of the branches with M results of “0” will be
w(M ) =
N!
pM q N −M
M !(N − M )!
(55)
where for convenience we have defined p = |α|2 and q = |β|2 +. The strategy
adopted is to bet whenever M/N ≥ p0 , where p0 x + (1 − p0 )y = ǫ, and the
expected utility of the strategy is then
Z ∞
X
1
N!
M N −M
√
EU =
p q
(px + qy) ≃ (px + qy)
dx exp(−x2 )
M !(N − M )!
π x0
M≥p0 N
(56)
p
where x0 = N/2pq(p0 − p).
Now, obviously if px + qy < 0 then this strategy has negative payoff no
matter what the values of ǫ and N — but observe that in such cases, the weight
of those branches in which the bet is accepted will be extremely small. Thus,
although the payoff will be negative it will be only very slightly negative, and
thus only trivially worse than not taking the bet; on the other hand if the
bet is objectively worth taking then (for large enough N ) the weight of those
branches in which it is taken will be close to unity. The strategy, then, seems
fairly well-supported.
8.3
A quantum Principal Principle
Lewis’s Principal Principle, briefly discussed in section 2.7, is claimed (with
some justice) to “capture all we know about chance” (Lewis 1980, p. 86), and it
is of some interest to know whether there is an Everett version of it (we have
already seen three scenarios — bets before a measurement, bets afterwards but
in ignorance of the result, and inference to the state — each of which makes a
somewhat different use of quantum probability, so there is certainly reason to
desire unification).
59
To state Lewis’s Principle (and our version) it will be useful to follow Lewis to
a more relaxed notion of subjective probability. Lewis’s subjective probabilities
— which he calls credences — are still in principle agent-relative, and still gain
their empirical significance by being the best analysis of an agent’s dispositions
to act. But we shall simply take it as read that they are defined over a very wide
class of situations, rather than carefully deriving this from decision-theoretic
axioms as hitherto.
To be specific, we will require an agent to have a credence function which
ranges over all propositions about the world (understood, if desired, in Lewis’s
sense as sets of possible worlds, but this is not required). Thus to any propositions A, B, there will exist numbers C(A), C(B) giving the agent’s credence
in the truth of those proposition, as well as a conditional credence C(A|B) =
C(A&B)/C(B) giving the credence in A given the known truth of B.
Actually, even this is not a sufficiently wide class over which to define credence. I wake up, sure of where I am sleeping but unsure of the time. I am not
ignorant of any facts about my world, only of my (temporal) location within it.
This suggests an extension of the credence function so that it ranges over properties (including, ‘the property of being in a world where proposition A holds’)
rather than just propositions; equivalently (in Lewis’s analysis) it ranges over
centred possible worlds — possible worlds with a preferred agent picked out —
rather than just over possible worlds simpliciter.
Lewis’s Principal Principle is then:
Let t be any time, and x a real number, and let X be the proposition
that the objective chance (at t) of A holding is x. Let E be any
proposition14 admissible at time t. Then
C(A|X&E) = x.
(57)
Lewis does not give an explicit analysis of ‘admissible’, but essentially the
reason for the restriction is to rule out already knowing what the result of the
chance event is, by time-travel, precognition or other occult methods. Such
events having been ruled out, the Principle basically says that credences equal
chances. For Lewis, past events aren’t chancy, so the Principle doesn’t directly
extend to credences about a past chance event (such as a measurement) when we
don’t know anything about the outcome; however, we can analyse these using
the Reflection Principle.
Here, then, is the Everettian version (which I will defend, below).
Everett Principal Principle (EPP): Let w(E) and w(A&E) be
real numbers; let A be any statement about a branch; let E any
conjunction of propositions, and statements about a branch. Let X
be the proposition that the weight of all branches in which E holds
is w(E), and that the weight of all the branches in which A and E
both hold is w(A&E). Then
C(A|X&E) = w(A&E)/w(E).
14 or
property, etc.
60
(58)
Let us consider some aspects of the EPP:
• The ‘statements about branches’ are to be understood along the lines of
the prototype: ‘in this branch, x is happening/ has happened’. As such
they are inherently indexical, and cannot be replaced by propositions. On
a technical level, each corresponds to some projector onto a coarse-graining
of the decoherence-preferred basis, or possibly a string of such projectors
b A ).
(we will denote such a string, for statement A, by P
• There is no need for a division of statements into ‘admissible’ and ‘inadmissible’. It is simply impossible for an agent to gain any inadmissible
information in the Everett interpretation: no amount of time-travelling,
precognition or the like can disguise the simple truth that there is no fact
of the matter about the future in the presence of quantum splitting, and
so no way to learn that fact.
• Despite the unity of the EPP, its credences have an inherently dualistic
nature. If A is determinate relative to the agent then C(A|X&E) is his
credence in the truth-value of some unknown, but determinately true or
false statement; if, however, A is not determinate (if, for instance, it refers
to the outcome of a quantum measurement to be performed in his future)
then his credence relates to the sort of subjective indeterminism discussed
in section 3.
• It is easy to see that EPP incorporates the Reflection Principle as used in
section 8.1.
• The EPP is perhaps more easily understood if we consider a limiting case:
suppose that E = X0 &B where X0 is the proposition that the Heisenberg
state of the universe is |ψi, and B is a statement about a branch. Since
X0 determines all the weights, it subsumes X (where X is as defined in
b A and P
bE
the EPP). If the projectors corresponding to A and E are P
respectively, then EPP gives
C(A|X0 &B) =
bB
b |ψi
hψ| A
.
b |ψi
hψ| A
(59)
The EPP, I think, gives us everything we might want from probabilities in
the Everett interpretation. Have we yet done enough to prove it? A proof along
the lines of this paper’s discussion might involve the following three steps:
1. Show that quantum-mechanical branching events are subjectively bestunderstood as situations of indeterminism, hence of ignorance about the
post-branching future.
2. Show that this ignorance is correctly analysed by means of the quantum
probability rule.
61
3. Extend this result to show that the quantum probability rule gives the
correct analysis of probability even when it concerns the objectively determinate (such as: what was the result of that measurement?).
(1) was argued for in section 3, following Saunders’ analysis. (2) was proved
from general axioms of decision theory in sections 4–7; the only loophole would
seem to be whether my quantum games are too stylised to count as models for
general quantum-mechanical branching events. And (3) can be argued for using
the Reflection Principle and asking how we would have selected our strategies
if we had made the selection before the original branching. (There is another
loophole here if the multiverse had multiple branches even at the beginning of
time, but this seems cosmologically implausible.)
The EPP, then, seems reasonably secure even if not completely proven. It is
satisfying to note that it at any rate appears to be on substantially more secure
foundations than any classical analysis of objective chance!
9
Objections; Replies; Conclusions
The main thrust of the argument of my paper, some of the obvious objections
and replies, and its conclusions, are summarised in the following dialogue. The
‘Everettian’ obviously speaks for the author, but at times so does the ‘Sceptic’
— his objections are not meant to be trivial and although I think they can be
answered there is probably much scope for further investigation.
Sceptic: What’s all this attention to ‘rationality’ in aid of? This is physics,
not psychology.
Everettian: Rationality is the only way in which the concept of ‘probability’
makes contact with the physical world.
Sceptic: But probability is just relative frequency in the limit.
Everettian: If by ‘limit’ you mean after an actual infinity of experiments,
then you’re welcome to define it that way if you like. But no-one can
actually carry out an actual infinity of experiments. In practice, we predict
probabilities from the relative frequencies of outcomes in finitely many
experiments. This isn’t automatically correct — probability theory itself
predicts a finite probability that it will fail — so we need some account of
why it’s rational to do it. Also, once we’ve got these “probabilities”, what
we actually do with them is use them as a guide to our expectations about
the future, and we need to know why that’s rational too. Arguably, if we
could do both we’d have a completely satisfactory analysis of probability.
Sceptic: Okay, so what do your rationality considerations give you in the Everett interpretation?
Everettian: Well, first we consider a quantum measurement, made on a known
state, and show that it’s rational for an agent to act as if the quantum
probability rule were true . . .
62
Sceptic: ‘As if’ it were true? Is it true or isn’t it?
Everettian: Okay, let’s backtrack a bit. The philosophy behind all this is
that probability is an operational notion: if some objective feature of
the physical world plugs into rationality considerations in just the way
that probability does, that feature is probability. Effectively we’re using
Lewis’s Principal Principle (or qualitative variants like Papineau’s Inferential and Decision-Theoretical links) as a criterion for what some chancecandidate has to do in order to be chance. (That’s how Lewis himself is
using the Principle: he requires that some account can be given of how
his best-systems analysis justifies the Principal Principle, and attacks rival
accounts of objective chance on the grounds that they couldn’t satisfy this
requirement.)
This sort of operationalism about the apparently fundamental is very much
in the spirit of the Everett interpretation, incidentally (the decoherencebased versions at any rate). See the author’s paper, Wallace 2001a, for
more information.
Sceptic: Okay, grant all this for the moment; where’s decision theory come in?
Everettian: Decision theory is a tool for decision-making under uncertainty. It
doesn’t introduce a primitive concept of (quantitative) probability at all,
incidentally — it just shows that rational decision-making requires us to
assign probabilities to events, values to consequences, and then use them
together to maximise expected utility.
Sceptic: If it’s about uncertainty, how can you apply it to a deterministic
theory?
Everettian: You have to buy into Saunders’ account of branching as a subjectively indeterministic process — then classical decision theory goes across
mutatis mutandis.
Sceptic: That account is surely open to attack?
Everettian: Well, you have to accept some reasonably strong philosophical
premises: Parfit’s criterion for personal identity, a fairly robust supervenience of the mental on the physical (probably not much short of out-andout functionalism), rejection of temporal becoming, and so forth. But
those are part and parcel of the Everett interpretation — reject them and
the interpretation comes apart as soon as we look at the preferred basis
problem, let alone at probability.
Sceptic: So assuming we accept Saunders’ account, what goes into the derivation?
Everettian: A standard set of decision-theoretic axioms such as Savage’s, minus most of the structure axioms (they are unnecessary because quantum
mechanics itself supplies all the structure), plus measurement neutrality:
63
the assumption that it doesn’t matter how a quantity is measured provided that it is measured.
Sceptic: That innocent-sounding assumption looks like a weak link.
Everettian: Well, yes. It seems possible to give it a pretty reasonable justification from the subjective-indeterministic viewpoint — and of course it’s
intuitively pretty obvious — but it’s obviously not something we can just
read off from the Savage axioms. And it’s doing a huge amount of work in
the proof — that’s mostly tacit in Deutsch’s paper, but the speed of the
proof from D0′ –D4′ or S0-S5 of the quantum probability rule shows how
powerful it is.
Sceptic: Don’t go appealing to intuitive obviousness here, either. The reason
we trust our measuring devices is that the operational part of quantum
mechanics — which is thoroughly tested — predicts that two different
devices have the same probability of a given outcome when they measure
the same observable on the same state. But you can’t help yourself to
this, since you’re trying to justify probability.
Everettian: Certainly we need to be careful about avoiding circularity in our
justification of measurement neutrality. But the probabilistic part of the
operational theory is not our only reason for accepting it — the major
reason is that a measurement device is clearly designed purely to register a
given eigenstate, in a deterministic fashion. An experimentalist who’s just
built a new device doesn’t deem it a z-spin-measurer because on testing
it gets the same results as some ‘canonical’ z-spin-measurer (though to be
sure, he might use some such device as a check.) He deems it so because
it’s designed such that if a given particle has a definite value of z-spin,
then the device will reliably tell us that value.
The property of measuring devices that leads to collapse (and thus, subjective indeterminism) is simply their property of magnifying superpositions
up to the macro level — and that property is constant across different
devices.
Sceptic: But for all we know some of the ‘irrelevant’ details of that superpositionmagnifying process are just those details which determine probabilities!
Everettian: Don’t fall into the trap of assuming that we’re after some new
physical law which tells us the probabilities. Ex hypothesi we know all the
laws already — what we’re looking for is a rationality principle.
Sceptic: Let’s move on: quite apart from technical worries about the proof, the
result seems conceptually far too weak to save the Everett interpretation.
It may tell me that it’s rational to bet that the frequencies on the next
zillion repetitions of the experiment will be what QM says, but what I
really need to know is why I should expect them to be!
64
Everettian: You expect them to be — that is, you think they’re highly probable. But in the account of probability that we’re using here, for something
to be highly probable just is for it to be rational to bet on it. If you
wanted to insist on some more fundamental understanding of probability,
or of what we should expect to happen, you should have left long ago.
Sceptic: Okay, point taken. But isn’t there something rather future-directed
about all this? We’re saying how we should act in the future, but we also
need to see how the theory is explanatory of the past — else we wouldn’t
have believed it in the first place.
Everettian: This is the point of the account (in section 3.3) of how a theory
is explanatory of data. The idea is essentially that an explanatory theory
makes the observed data highly probable15 (I mean to say, it makes it
rational to bet on its occurrence). Thus there’s an essentially futurelooking aspect even to our assessment of the theory’s relevance for past
data: we have to imagine ourselves just before the data were collected,
and ask what we should expect.
Sceptic: So far, we’ve only been skirmishing. We’ve been discussing problems
within the author’s program, but now let’s move on to reasons why no such
program could work. To begin: the theory is realist about all branches,
so it accepts that there are many branches in which the frequencies are
nowhere near those predicted by the quantum algorithm.
Everettian: Yes, but this isn’t one of them.
Sceptic: How can you tell?
Everettian: I retract slightly: it’s rational to assume that this isn’t one of
them (see section 8 for the proof).
Sceptic: But it might be.
Everettian: Sure, and Elvis might have shot JFK, but it would be rational
to assume neither to be true. In fact it’s overwhelmingly more rational
to bet on Elvis shooting JFK than to bet that this is a wildly anomalous
branch. This really isn’t any different from ordinary probability: there’s
always some chance of anomalous statistics.
Sceptic: Everettians often trot out that last response. The difference is, in
classical probability the anomalous branches aren’t actualised.
Everettian: What you mean is, they probably aren’t actualised. And here’s
where we can give a unified account of classical and quantum cases: what
we mean by ‘they probably aren’t actualised’ is that it’s rational to assume
that they won’t be actualised.
15 Technically, it only makes the observed data a member of some ‘typical set’ of data which
is highly probable; cf. footnote on page 20.
65
Sceptic: But in the quantum case — and not the classical — the anomalous
branches actually exist. In these branches, scientific communities come
up with wildly different theories to yours.
Everettian: These other theories are wrong, though.
Sceptic: How can you have the confidence to say that? And don’t say, ‘it’s
rational to assume it.’ They’ve also been arrived at by a rational process;
presumably they’d find it rational to reject your conclusion. Doesn’t this
undermine faith in your own argument?
Everettian: Not really. It’s not problematic for my theory to predict that
other people will reject it for rational reasons, provided that it also predicts
that I’m not one of them. There’s no neutral place to stand in this game.
What’s more, there’s nothing particularly quantum about this supposed
paradox. Suppose Alf experiences a truly spooky coincidence, the sort
of fluke that would only have a one-in-a-billion chance of occurring in his
lifetime — unless ghosts exist, in which case it’s not surprising at all. Then
it would be rational for him to drastically increase his credence in ghosts,
unless he already thought the chance of their existence far lower than one
in a billion. Yet on a planet with six billion inhabitants, there will be
several Alfs even if ghosts do not exist. I may be sure that they’re wrong
(or at least, overwhelmingly likely to be wrong) about ghosts, but that
doesn’t force me to conclude that they’re irrational — and the consistency
of the situation is guaranteed by the prediction that it’s irrational for me
to expect to be that person.16
Sceptic: The strongest objection has been saved till last. If I accept Parfit’s
account of what matters in personal identity, then I care about my future
descendants because of the structural and causal relation which I bear to
them. Why, then, should I give a damn about their relative weights?
Everettian: Because it would be irrational not to.
Sceptic: It is unclear who has the burden of proof: my argument is precisely
that it would be irrational to care about the weights.
Everettian: Your argument is based upon the objective-deterministic viewpoint, which (it has been argued) is not the right view to assess rationality
16 Let A be “a one-in-a-billion spooky coincidence”, and B be “ghosts exist”; assume that
Pr(B) ≪ 1 (uncontroversially, I hope!), that Pr(A|¬B) = 10−9 , and that Pr(A|B) is unity.
Bayes’ Theorem tells us that
Pr(B|A) ≃
Pr(B)
10−9 + Pr(B)
(60)
which is close to unity even if the prior probability of B is one in a hundred million — and
how many of us are that sure of our twenty-first century worldview? (I am quite, quite sure
that ghosts don’t exist — sure enough to bet my life on it, far more certain of it than I am of
living until tomorrow — yet I’m not that sure. 108 is a big number.) I am grateful to Simon
Saunders for the inspiration behind this example.
66
considerations in Everett. From the subjective-indeterministic view it is a
theorem that it is rational to weigh options in proportion to their quantum
weights, and your approach violates that.
Sceptic: But my objection cuts deeper than that: if (as I claim) it is obviously rational to care equally about both descendants, and if subjective
expectation is supposed somehow to supervene on the degree to which I
do care for them, surely equiprobability has to go into decision theory as
a premise. If its denial is a theorem, then the system is inconsistent; so
much the worse, then, for Everett.
Everettian: That would be fine, if you could justify the supervenience of expectation on “caring for”. But no such assumption went into Saunders’
argument. All he uses is the fact that if I have multiple descendants then
I should expect to be exactly one of them; this is enough to apply decision
theory.
Sceptic: Suppose I accept this: even so, once I accept the Everett interpretation then I would be at liberty to use the objective-deterministic viewpoint. From its perspective, I could then argue that the probability rule
should be revised. If this contravened the Savage axioms, so be it: not
even axioms of pure rationality are immune to revision.
Everettian: Maybe so, though the rest of us needn’t join you in that revision
if we’re instead willing to give up your equiprobability argument upon
realising that it clashes with the Savage axioms: apparently reasonable
strategies are certainly not immune to revision if it is shown that they
contravene obvious principles of rationality, and the Savage axioms are
surely more obvious than the correct strategy to take in a case of fission!
I guess there’s a transcendental argument why, if you’re sure you’d adopt
the equiprobability rule upon becoming an Everettian, you’d be irrational
to become one. But if I were sure that I found the de Broglie-Bohm theory
so unaesthetic that I’d lapse into depression and suicide were I to come to
believe it, I would be rational to avoid believing it . . . but that wouldn’t
really bear on its truth.
Incidentally, one possible reason why you should reject your equiprobability rule on its own terms is that it assumes that the decoherence process consists of a discrete number of splittings, each into finitely many
branches. In reality this is implausible: branching isn’t as precisely defined as all that, and there may well be infinitely or even continuously
many branches.
Sceptic: The Bekenstein bound shows that Hilbert space is finite-dimensional.
Everettian: I’d prefer to leave quantum gravity out of it. It’s likely to change
our theories enough that effects on the dimension of Hilbert space will be
the least of our worries — but we don’t yet have any clear idea how, and the
67
Bekenstein bound is at present just a plausible bit of speculation. Come
to that, the decision-theoretic approach doesn’t tell us how to interpret
the non-unitary evolution occurring in black hole evaporation, and that’s
about as secure (or otherwise) as the Bekenstein bound.
Sceptic: Let us call a truce for now: if your account really is valid, what do
you conclude from it?
Everettian: Most importantly, that we can vindicate Deutsch’s claim: that
applying decision theory to quantum mechanics is enough to solve the
probability problem in the Everett interpretation. Decision theory provides a framework in which we can understand what is involved in deducing quantitative probabilities for quantum branching, and then shows us
that this can be done satisfactorily even when questionable assumptions
like additivity are abandoned. Furthermore, the relevant links between
quantum probability and non-probabilistic facts can then be satisfactorily
established.
Just as interesting are the implications of quantum theory for decision
theory and the general philosophy of probability. On the technical side,
it is noteworthy that the structure axioms required throughout classical
decision theory can be very substantially weakened. To be sure, this is
only because the mathematical structure of the physical theory (i. e. , QM)
in which the decision problem is posed is so rich, but it seems far more
satisfactory to have a richly structured physical theory (whose structure
is clearly required on directly empirical grounds and in any case is ontologically on a par with any other postulate of physical theorising) rather
than introduce axioms governing rationality which are not self-evident and
which fairly clearly are introduced purely to guarantee a representation
theorem.
On a more conceptual level, (Everettian) quantum mechanics seems to
provide a novel route by which the concept of objective chance can be
introduced. Everettian objective chances supervene entirely on particular
matters of fact (Lewis’s Humean requirement for chance), but the Everettian account is an improvement on Lewis’s in that chances supervene
on local facts only, rather than on the pattern of all facts up to the time of
the chance event. Furthermore, an account of how these chances connect
with credence is available that is at least as secure as the frequency-based
account — indeed, though we do not have a full derivation of the Everett
Principal Principle, we have come close.
Sceptic: What would be a simple way of seeing how all this is possible: how
quantum mechanics can have these consequences for decision theory, and
how the derivation of the quantum probability rule was possible in the
first place?
Everettian: It’s long been recognised that the most fruitful guides to allocation of probability have been frequencies and symmetries, but the latter
68
has always been somewhat suspect, and it is easy to see why: how are we
to choose which symmetries are respected by the chances? Appeal to the
symmetries of the physical laws seems the obvious method, but obviously
this just begs the question if those laws are probabilistic. Even for deterministic laws, though, the situation is problematic: for if the situation
is completely symmetric between two outcomes, how is it that one outcome happens rather than the other? In classical mechanics, for instance,
knowledge of the exact microstate of a flipped coin breaks the symmetry of
that coin and tells us with certainty which side it will land. The symmetry
only enters because we assume the coin’s microstate to be distributed randomly with 50% probability of leading to each result, but this introduces
probability in advance rather than deriving it from symmetry.17
In the Everett interpretation, this circle is squared: it is not the case
that one or the other outcome will occur; from a God’s-eye view, both
do. This allows us to apply symmetry-based reasoning to equal-amplitude
branching — the core of the proofs in sections 5–6 — and deduce that the
branches are equiprobable.
In a sense, then, the Everett interpretation reverses the primacy of frequency over symmetry: the frequency of outcomes is an excellent guide
to the symmetry of the state being measured, but ultimately it is the
symmetries which dictate which events are equiprobable.
Acknowledgements
For valuable discussions, I am indebted to Hannah Barlow, Katherine Brading,
Harvey Brown, Jeremy Butterfield, Adam Elga, Chris Fuchs, Hilary Greaves,
Adrian Kent, Chris Timpson, Wojciech Zurek, to all those at the 2002 OxfordPrinceton philosophy of physics workshop, and especially to Simon Saunders and
David Deutsch. Jeremy Butterfield and Simon Saunders also made detailed and
helpful comments on an earlier draft of this paper.
Appendix: act and consequence additivity
Theorem A. 1 Suppose that we have some set C, together with a binary addition operation + that is associative, commutative and has identity 0. Let ≻ be
a weak asymmetric order on C, satisfying
1. y ≻ z if and only if y + x ≻ z + x for some x;
17 This
is perhaps mildly unfair: in statistical mechanics we make a general postulate that
systems are randomly distributed in the accessible region of phase space according to the
Liouville measure, and then count the symmetry of the coin as telling us that the regions of
phase space leading to the two outcomes are of equal measure. Nonetheless the underlying
argument still contains a strong initial admixture of probability.
69
2. Whenever y ≻ z ≻ 0, there exists some integer n such that ny ≻ (and
conversely for y ≺ z ≺ 0);
3. Whenever y ≺ 0 and x ≻ 0, there exists some integer n such that y + nx ≻
0.
4. For any c1 , c2 where c1 ≻ c2 , and for any c, there exist integers m, n such
that nc1 ≻ mc ≻ nc2 .
Then there exists a function V : C → ℜ, unique up to a multiplicative constant,
such that (a) V(c1 ) > V(c2 ) if and only if c1 ≻ c2 and (b) V(c1 + c2 ) = V(c1 ) +
V(c2 ).
Proof: We can suppose without loss of generality that there is at least one
element x ≻ 0 (if not, but there is at least one x ≺ 0, define a new ordering
≻′ =≺, construct its value function, and take its negative; if for all x, x ≃ 0,
then just put V(x) = 0.)
For all other elements y ≻ 0, define the set Val(y) = {m/n | ny ≻ mx}.
Lemma 1 Val(y) is closed below: if m′ /n′ < m/n and m/n ∈ Val(y), then
m′ /n′ ∈ Val(y).
Now, suppose m′ /n′ < m/n (so, m′ n < mn′ ).
m/n ∈ Val(y) =⇒ ny ≻ mx
=⇒ nn′ y ≻ mn′ x
=⇒ nn′ y ≻ m′ nx
=⇒ n′ y ≻ m′ x
=⇒ m′ /n′ ∈ Val(y).2
Lemma 2 Val(y) has no greatest element.
Suppose m/n ∈ Val(y). We have to show that there exist m′ , n′ such that
m′ /n′ ∈ Val(y) and m′ /n′ > m/n. We have ny ≻ mx, so by postulate 4 there
exist integers a, b such that nay ≻ bx ≻ max. Put m′ = b, n′ = na. Then since
b > ma, m′ /n′ > m/n. 2
Further, Val(y) is non-empty, since it contains 0; it does not contain all
rational numbers, by postulate 3.
But any subset of rational numbers which is non-empty, not equal to the set
of all rationals, closed below, and without greatest element is a Dedekind cut,
and thus can be regarded as a real number: we thus take V(y) to be the real
associated with the Dedekind cut Val(y).
To establish that V has the right properties, suppose first that V(y) >> V(z).
Then there exist m, n such that ny ≻ mx but ny mx. Hence ny ≻ nz, so
y ≻ z.
70
The converse requires another use of postulate 4 (to rule out the possibility
that z is infinitesimally less valuable than y). Suppose y ≻ z, then there exist
m and n such that ny ≻ mx ≻ nz. Hence, m/n is in Val(y) but not Val(z), so
V(y) > V(z).
This completes the construction of the value function for all y ≻ 0. For y ≺ 0
we define Val(y) = {m/n | 0 ≻ ny + mx}, and prove in an essentially identical
way that this is also a Dedekind cut. 2
Theorem A. 2 Let S = {s1 , . . . , sn } be a finite set (of states); let C be a
set of consequences on which there exists an addition operation. Let A, the
act space, be defined as the set of functions from S to C (to be thought of as
bets on which state will occur). Define addition of acts in the obvious way:
(P1 + P2 )(si ) = P1 (si ) + P2 (si ). Now suppose that there exists a preference
order ≻ on A, satisfying
1. Additivity: ≻ can be represented by an additive value function V on A
(equivalently, ≻ obeys the postulates of Theorem A.1). (We write V(x)
for the value of the constant act P(s) = x for all s.)
2. Dominance: if V[P(si )] ≥ V[P ′ (si )] for all i, then V(P) ≥ V(P ′ ).
P
Then there exist positive P
real numbers p1 , . . . , pn , satisfying i pi = 1, such
that for any act P, V(P) = i pi V[P(si )].
Proof: For any consequence x ≻ 0 and any state index i, define Px,i = xδi,j .
The sum over i of all the acts Px,i is just the constant act P(s) = x, hence
X
V(Px,i ) = V(x).
i
P
If we define pi (x) = V(Px,i )/V(x), we have i pi (x) = 1; by the Dominance
assumption, each V(Px,i ) ≥ 0, so each pi (x) ≥ 0. Our remaining task is to prove
that the pi are independent of x.
Suppose y ≻ x. (We shall not give the proof for x ≻ y ≻ 0 or y ≺ 0, as it
is essentially identical.) Let αj = mj /nj be an increasing sequence of rational
numbers such that limj→∞ αj = V(y)/V(x). By Dominance and Additivity, it
follows that
mj V(Py,i ) = V(Pmj y,i ) ≥ V(Pnj x,i ) = nj pi (x)V(x),
and hence V(Py,i ) ≥ αpi (x)V(y). Similar consideration of a decreasing sequence
gives us V(Py,i ) ≤ pi (x)V(y). Between these two inequalities we can read off
pi (y) = pi (x), and the result is proved.
References
Albert, D. and B. Loewer (1988). Interpreting the Many Worlds Interpretation. Synthese 77, 195–213.
71
Barnum, H., C. M. Caves, J. Finkelstein, C. A. Fuchs, and R. Schack
(2000). Quantum Probability from Decision Theory? Proceedings of
the Royal Society of London A456, 1175–1182. Available online at
http://www.arXiv.org/abs/quant-ph/9907024.
Bohm, D. (1952). A Suggested Interpretation of Quantum Theory in Terms
of “Hidden” Variables. Physical Review 85, 166–193.
de Finetti, B. (1974). Theory of Probability. New York: John Wiley and Sons,
Inc,.
Dennett, D. C. (1987). The intentional stance. Cambridge, Mass.: MIT Press.
Dennett, D. C. (1991). Real Patterns. Journal of Philosophy 87, 27–51.
Reprinted in Brainchildren, D. Dennett, (London: Penguin 1998) pp.
95–120.
Deutsch, D. (1985). Quantum Theory as a Universal Physical Theory. International Journal of Theoretical Physics 24 (1), 1–41.
Deutsch, D. (1996). Comment on Lockwood. British Journal for the Philosophy of Science 47, 222–228.
Deutsch, D. (1999). Quantum Theory of Probability and Decisions. Proceedings of the Royal Society of London A455, 3129–3137. Available online at
http://www.arxiv.org/abs/quant-ph/9906015.
Deutsch, D. (2001). The Structure of the Multiverse. Available online at
http://xxx.arXiv.org/abs/quant-ph/0104033.
DeWitt, B. (1970). Quantum Mechanics and Reality. Physics Today 23 (9),
30–35. Reprinted in (DeWitt and Graham 1973).
DeWitt, B. and N. Graham (Eds.) (1973). The many-worlds interpretation of
quantum mechanics. Princeton: Princeton University Press.
Donald, M. (1997). On Many-Minds Interpretations of Quantum Theory.
Available online at http://www.arxiv.org/abs/quant-ph/9703008.
Elga, A. (2000). Self-locating belief and the sleeping beauty problem. Analysis 60 (2), 143–147.
Everett III, H. (1957). Relative State Formulation of Quantum Mechanics.
Review of Modern Physics 29, 454–462. Reprinted in DeWitt and Graham
(1973).
Fishburn, P. C. (1981). Subjective expected utility: A review of normative
theories. Theory and Decision 13, 139–99.
Gärdenfors, P. and N.-E. Sahlin (Eds.) (1988). Decision, Probability and Utility: Selected Readings. Cambridge: Cambridge University Press.
Gleason, A. (1957). Measures on the closed subspaces of a hilbert space.
Journal of Mathematics and Mechanics 6, 885–894.
Holland, P. (1993). The Quantum Theory of Motion. Cambridge: Cambridge
University Press.
72
Howson, C. and P. Urbach (1989). Scientific Reasoning: the Bayesian Approach (2nd edition). Chicago and La Salle, Illinois: Open Court Publishing Company.
Joyce, J. N. (1999). The Foundations of Causal Decision Theory. Cambridge:
Cambridge University Press.
Kaplan, M. (1983). Decision theory as philosophy. Philosophy of Science 50,
549–577.
Lewis, D. (1980). A subjectivist’s guide to objective chance. In R. C. Jeffrey
(Ed.), Studies in Inductive Logic and Probability, Volume II. Berkeley:
University of California Press. Reprinted in David Lewis, Philosophical
Papers, Volume II (Oxford University Press, Oxford, 1986).
Lewis, D. (1994). Chance and credence: Humean supervenience debugged.
Mind 103, 473–90.
Lockwood, M. (1989). Mind, Brain and the Quantum: the compound ‘I’.
Oxford: Blackwell Publishers.
Lockwood, M. (1996). ‘Many Minds’ Interpretations of Quantum Mechanics.
British Journal for the Philosophy of Science 47, 159–188.
Mellor, D. H. (1971). The Matter of Chance. Cambridge: Cambridge University Press.
Papineau, D. (1996). Many Minds are No Worse than One. British Journal
for the Philosophy of Science 47, 233–241.
Parfit, D. (1984). Reasons and Persons. Oxford: Oxford University Press.
Quine, W. v. O. (1960). Word and Object. Cambridge, Mass.: MIT Press.
Ramsey, F. P. (1931). Truth and probability. In The Foundations of Mathematics and Other Logical Essays, R. B. Braithwaite (ed. ) (Routledge and
Kegan Paul, London) pp. 156–198; reprinted in (Gärdenfors and Sahlin
1988), pp. 19–47.
Redhead, M. (1987). Incompleteness, Nonlocality and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics. Oxford: Oxford University Press.
Saunders, S. (1995). Time, Decoherence and Quantum Mechanics. Synthese 102, 235–266.
Saunders, S. (1997). Naturalizing Metaphysics. The Monist 80 (1), 44–69.
Saunders, S. (1998). Time, Quantum Mechanics, and Probability. Synthese 114, 373–404.
Saunders, S. (2002). Derivation of the Born Rule from Operational Assumptions. Forthcoming; available online at http:/www.arXiv.org.
Savage, L. J. (1972). The foundations of statistics (2nd edition ed.). New
York: Dover.
73
Sudbery, A. (2000). Why am I me? and why is my world so classical? Available online at http://www.arxiv.org/abs/quant-ph/0011084.
Vaidman, L. (1998). On Schizophrenic Experiences of the Neutron or Why
We Should Believe in the Many-Worlds Interpretation of Quantum Theory. International Studies in Philosophy of Science 12, 245–261. Available
online at http://www/arxiv.org/abs/quant-ph/9609006.
Vaidman, L. (2001). The Many-Worlds Interpretation of Quantum Mechanics. To appear in the Stanford Encyclopedia of Philosophy; temporarily
available online at http://www.tau.ac.il∼vaidman/mwi/mwst1.html.
van Fraassen, B. (1984). Belief and the will. Journal of Philosophy LXXXI,
235–256.
von Neumann, J. and O. Morgenstern (1947). Theory of Games and Economic
Behaviour (2nd ed.). Princeton: Princeton University Press.
Wallace, D. (2001a). Everett and Structure. Forthcoming in Studies
in the History and Philosophy of Modern Physics; available online
at http://xxx.arXiv.org/abs/quant-ph/0107144 or from http://philsciarchive.pitt.edu.
Wallace, D. (2001b). Worlds in the Everett Interpretation. Forthcoming in
Studies in the History and Philosophy of Modern Physics; available online
at http://www.arxiv.org/abs/quant-ph/0103092 or from http://philsciarchive.pitt.edu.
Zurek, W. H. (1991). Decoherence and the Transition from Quantum to Classical. Physics Today 44, 36–44.
Zurek, W. H. (1998). Decoherence, Einselection and the Existential Interpretation (The Rough Guide). Philosophical Transactions of the Royal
Society of London A356, 1793–1820.
Zurek, W. H. (2001). Decoherence, Einselection, and the Quantum Origins of the Classical. Available online at http://xxx.arXiv.org/abs/quantph/0105127; to appear in Reviews of Modern Physics.
74